{"id":1203,"date":"2021-06-30T17:02:09","date_gmt":"2021-06-30T17:02:09","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/problem-set-determining-volumes-by-slicing\/"},"modified":"2021-12-09T00:28:25","modified_gmt":"2021-12-09T00:28:25","slug":"problem-set-determining-volumes-by-slicing","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/problem-set-determining-volumes-by-slicing\/","title":{"raw":"Problem Set: Determining Volumes by Slicing","rendered":"Problem Set: Determining Volumes by Slicing"},"content":{"raw":"
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1.\u00a0<\/strong>Derive the formula for the volume of a sphere using the slicing method.<\/p>\r\n\r\n<\/div>\r\n

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2.\u00a0<\/strong>Use the slicing method to derive the formula for the volume of a cone.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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3.\u00a0<\/strong>Use the slicing method to derive the formula for the volume of a tetrahedron with side length [latex]a.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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4.\u00a0<\/strong>Use the disk method to derive the formula for the volume of a trapezoidal cylinder.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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5.\u00a0<\/strong>Explain when you would use the disk method versus the washer method. When are they interchangeable?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (6-10), draw a typical slice and find the volume using the slicing method for the given volume.<\/p>\r\n\r\n

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6.\u00a0<\/strong>A pyramid with height 6 units and square base of side 2 units, as pictured here.<\/p>\r\n\"This<\/span>\r\n[reveal-answer q=\"fs-id1167793879238\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793879238\"]\r\n\r\n8 units3\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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7.\u00a0<\/strong>A pyramid with height 4 units and a rectangular base with length 2 units and width 3 units, as pictured here.<\/p>\r\n\"This<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n

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8.\u00a0<\/strong>A tetrahedron with a base side of 4 units, as seen here.<\/p>\r\n\"This<\/span>\r\n[reveal-answer q=\"fs-id1167794026478\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794026478\"]\r\n\r\n[latex]\\frac{32}{3\\sqrt{2}}[\/latex] units3\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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9.\u00a0<\/strong>A pyramid with height 5 units, and an isosceles triangular base with lengths of 6 units and 8 units, as seen here.<\/p>\r\n\"This<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n

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10.\u00a0<\/strong>A cone of radius [latex]r[\/latex] and height [latex]h[\/latex] has a smaller cone of radius [latex]r\\text{\/}2[\/latex] and height [latex]h\\text{\/}2[\/latex] removed from the top, as seen here. The resulting solid is called a frustum<\/em>.<\/p>\r\n\"This<\/span>\r\n[reveal-answer q=\"fs-id1167793932196\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793932196\"]\r\n\r\n[latex]\\frac{7\\pi }{12}h{r}^{2}[\/latex] units3\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (11-16), draw an outline of the solid and find the volume using the slicing method.<\/p>\r\n\r\n

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11.\u00a0<\/strong>The base is a circle of radius [latex]a.[\/latex] The slices perpendicular to the base are squares.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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12.\u00a0<\/strong>The base is a triangle with vertices [latex](0,0),(1,0),[\/latex] and [latex](0,1).[\/latex] Slices perpendicular to the xy<\/em>-plane are semicircles.<\/p>\r\n[reveal-answer q=\"fs-id1167793929065\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793929065\"]\r\n\"This<\/span>\r\n[latex]\\frac{\\pi }{24}[\/latex] units3<\/sup>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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13.\u00a0<\/strong>The base is the region under the parabola [latex]y=1-{x}^{2}[\/latex] in the first quadrant. Slices perpendicular to the xy<\/em>-plane are squares.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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14.\u00a0<\/strong>The base is the region under the parabola [latex]y=1-{x}^{2}[\/latex] and above the [latex]x\\text{-axis}\\text{.}[\/latex] Slices perpendicular to the [latex]y\\text{-axis}[\/latex] are squares.<\/p>\r\n[reveal-answer q=\"fs-id1167794122174\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794122174\"]\"This<\/span>\r\n2 units3<\/sup>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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15.\u00a0<\/strong>The base is the region enclosed by [latex]y={x}^{2}[\/latex] and [latex]y=9.[\/latex] Slices perpendicular to the [latex]x[\/latex]-axis are right isosceles triangles.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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16.\u00a0<\/strong>The base is the area between [latex]y=x[\/latex] and [latex]y={x}^{2}.[\/latex] Slices perpendicular to the [latex]x[\/latex]-axis are semicircles.<\/p>\r\n[reveal-answer q=\"fs-id1167794330330\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794330330\"]\r\n\"This<\/span>\r\n[latex]\\frac{\\pi }{240}[\/latex] units3<\/sup>[\/hidden-answer]\r\n\r\n<\/div>\r\nFor the following exercises (17-24), draw the region bounded by the curves. Then, use the disk method to find the volume when the region is rotated around the [latex]x[\/latex]-axis.<\/span>\r\n\r\n<\/div>\r\n

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17.\u00a0<\/strong>[latex]x+y=8,x=0,\\text{ and }y=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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18.\u00a0<\/strong>[latex]y=2{x}^{2},x=0,x=4,\\text{ and }y=0[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793566161\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793566161\"]\"This<\/span>\r\n[latex]\\frac{4096\\pi }{5}[\/latex] units3<\/sup>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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19.\u00a0<\/strong>[latex]y={e}^{x}+1,x=0,x=1,\\text{ and }y=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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20.\u00a0<\/strong>[latex]y={x}^{4},x=0,\\text{ and }y=1[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167794327644\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794327644\"]\"This<\/span>\r\n[latex]\\frac{8\\pi }{9}[\/latex] units3<\/sup>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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21.\u00a0<\/strong>[latex]y=\\sqrt{x},x=0,x=4,\\text{ and }y=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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22.\u00a0<\/strong>[latex]y= \\sin x,y= \\cos x,\\text{ and }x=0[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793269330\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793269330\"]\"This<\/span>\r\n[latex]\\frac{\\pi }{2}[\/latex] units3<\/sup>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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23.\u00a0<\/strong>[latex]y=\\frac{1}{x},x=2,\\text{ and }y=3[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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24.\u00a0<\/strong>[latex]{x}^{2}-{y}^{2}=9\\text{ and }x+y=9,y=0\\text{ and }x=0[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793928428\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793928428\"]\"This<\/span>\r\n[latex]207\\pi [\/latex] units3<\/sup>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (25-32), draw the region bounded by the curves. Then, find the volume when the region is rotated around the [latex]y[\/latex]-axis.<\/p>\r\n\r\n

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25.\u00a0<\/strong>[latex]y=4-\\frac{1}{2}x,x=0,\\text{ and }y=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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26.\u00a0<\/strong>[latex]y=2{x}^{3},x=0,x=1,\\text{ and }y=0[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793626588\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793626588\"]\"This<\/span>\r\n[latex]\\frac{4\\pi }{5}[\/latex] units3<\/sup>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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27.\u00a0<\/strong>[latex]y=3{x}^{2},x=0,\\text{ and }y=3[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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28.\u00a0<\/strong>[latex]y=\\sqrt{4-{x}^{2}},y=0,\\text{ and }x=0[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793541208\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793541208\"]\"This<\/span>\r\n[latex]\\frac{16\\pi }{3}[\/latex] units3<\/sup>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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29.\u00a0<\/strong>[latex]y=\\frac{1}{\\sqrt{x+1}},x=0,\\text{ and }x=3[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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30.\u00a0<\/strong>[latex]x= \\sec (y)\\text{ and }y=\\frac{\\pi }{4},y=0\\text{ and }x=0[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167794324574\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794324574\"]\"This<\/span>\r\n[latex]\\pi [\/latex] units3<\/sup>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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\r\n\r\n31.\u00a0<\/strong>[latex]y=\\frac{1}{x+1},x=0,\\text{ and }x=2[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n
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32.\u00a0<\/strong>[latex]y=4-x,y=x,\\text{ and }x=0[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793455016\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793455016\"]\"This<\/span>\r\n[latex]\\frac{16\\pi }{3}[\/latex] units3<\/sup>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (33-40), draw the region bounded by the curves. Then, find the volume when the region is rotated around the [latex]x[\/latex]-axis.<\/p>\r\n\r\n

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33.<\/strong> [latex]y=x+2,y=x+6,x=0,\\text{ and }x=5[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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34.\u00a0<\/strong>[latex]y={x}^{2}\\text{ and }y=x+2[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793316072\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793316072\"]\"This<\/span>\r\n[latex]\\frac{72\\pi }{5}[\/latex] units3<\/sup>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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35.\u00a0<\/strong>[latex]{x}^{2}={y}^{3}\\text{ and }{x}^{3}={y}^{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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36.\u00a0<\/strong>[latex]y=4-{x}^{2}\\text{ and }y=2-x[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793446635\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793446635\"]\"This<\/span>\r\n[latex]\\frac{108\\pi }{5}[\/latex] units3<\/sup>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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37. [T]<\/strong> [latex]y= \\cos x,y={e}^{\\text{\u2212}x},x=0,\\text{ and }x=1.2927[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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38.\u00a0<\/strong>[latex]y=\\sqrt{x}\\text{ and }y={x}^{2}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793637336\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793637336\"]\"This<\/span>\r\n[latex]\\frac{3\\pi }{10}[\/latex] units3<\/sup>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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39.\u00a0<\/strong>[latex]y= \\sin x\\text{,}y=5 \\sin x,x=0\\text{ and }x=\\pi [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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40.\u00a0<\/strong>[latex]y=\\sqrt{1+{x}^{2}}\\text{ and }y=\\sqrt{4-{x}^{2}}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793414076\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793414076\"]\"This<\/span>\r\n[latex]2\\sqrt{6}\\pi [\/latex] units3<\/sup>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (41-45), draw the region bounded by the curves. Then, use the washer method to find the volume when the region is revolved around the [latex]y[\/latex]-axis.<\/p>\r\n\r\n

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41.\u00a0<\/strong>[latex]y=\\sqrt{x},x=4,\\text{ and }y=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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42.\u00a0<\/strong>[latex]y=x+2,y=2x-1,\\text{ and }x=0[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167794140630\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794140630\"]\"This<\/span>\r\n[latex]9\\pi [\/latex] units3<\/sup>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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43.\u00a0<\/strong>[latex]y=\\sqrt[3]{x}\\text{ and }y={x}^{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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44.\u00a0<\/strong>[latex]x={e}^{2y},x={y}^{2},y=0,\\text{ and }y=\\text{ln}(2)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167794094540\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794094540\"]\"This<\/span>\r\n[latex]\\frac{\\pi }{20}(75-4{\\text{ln}}^{5}(2))[\/latex] units3<\/sup>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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45.\u00a0<\/strong>[latex]x=\\sqrt{9-{y}^{2}},x={e}^{\\text{\u2212}y},y=0,\\text{ and }y=3[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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46.\u00a0<\/strong>Yogurt containers can be shaped like frustums. Rotate the line [latex]y=\\frac{1}{m}x[\/latex] around the [latex]y[\/latex]-axis to find the volume between [latex]y=a\\text{ and }y=b.[\/latex]<\/p>\r\n\"This<\/span>\r\n[reveal-answer q=\"fs-id1167794329489\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794329489\"]\r\n\r\n[latex]\\frac{{m}^{2}\\pi }{3}({b}^{3}-{a}^{3})[\/latex] units3\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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47.\u00a0<\/strong>Rotate the ellipse [latex]({x}^{2}\\text{\/}{a}^{2})+({y}^{2}\\text{\/}{b}^{2})=1[\/latex] around the [latex]x[\/latex]-axis to approximate the volume of a football, as seen here.<\/p>\r\n\"This<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n

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48.\u00a0<\/strong>Rotate the ellipse [latex]({x}^{2}\\text{\/}{a}^{2})+({y}^{2}\\text{\/}{b}^{2})=1[\/latex] around the [latex]y[\/latex]-axis to approximate the volume of a football.<\/p>\r\n[reveal-answer q=\"fs-id1167793521343\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793521343\"]\r\n

[latex]\\frac{4{a}^{2}b\\pi }{3}[\/latex] units3<\/sup><\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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49.\u00a0<\/strong>A better approximation of the volume of a football is given by the solid that comes from rotating [latex]y= \\sin x[\/latex] around the [latex]x[\/latex]-axis from [latex]x=0[\/latex] to [latex]x=\\pi .[\/latex] What is the volume of this football approximation, as seen here?<\/p>\r\n\"This<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n

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50.\u00a0<\/strong>What is the volume of the Bundt cake that comes from rotating [latex]y= \\sin x[\/latex] around the [latex]y[\/latex]-axis from [latex]x=0[\/latex] to [latex]x=\\pi ?[\/latex]<\/p>\r\n\"This<\/span>\r\n[reveal-answer q=\"fs-id1167794095272\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794095272\"]\r\n\r\n[latex]2{\\pi }^{2}[\/latex] units3\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\nFor the following exercises (51-56), find the volume of the solid described.\r\n

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51. <\/strong>The base is the region between [latex]y=x[\/latex] and [latex]y={x}^{2}.[\/latex] Slices perpendicular to the [latex]x[\/latex]-axis are semicircles.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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52.\u00a0<\/strong>The base is the region enclosed by the generic ellipse [latex]({x}^{2}\\text{\/}{a}^{2})+({y}^{2}\\text{\/}{b}^{2})=1.[\/latex] Slices perpendicular to the [latex]x[\/latex]-axis are semicircles.<\/p>\r\n[reveal-answer q=\"fs-id1167793705258\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793705258\"]\r\n

[latex]\\frac{2a{b}^{2}\\pi }{3}[\/latex] units3<\/sup><\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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53.\u00a0<\/strong>Bore a hole of radius [latex]a[\/latex] down the axis of a right cone and through the base of radius [latex]b,[\/latex] as seen here.<\/p>\r\n\"This<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n

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54.\u00a0<\/strong>Find the volume common to two spheres of radius [latex]r[\/latex] with centers that are [latex]2h[\/latex] apart, as shown here.<\/p>\r\n\"This<\/span>\r\n[reveal-answer q=\"fs-id1167793544882\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793544882\"]\r\n\r\n[latex]\\frac{\\pi }{12}{(r+h)}^{2}(6r-h)[\/latex] units3\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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55.\u00a0<\/strong>Find the volume of a spherical cap of height [latex]h[\/latex] and radius [latex]r[\/latex] where [latex]h<r,[\/latex] as seen here.<\/p>\r\n\"This<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n

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56.\u00a0<\/strong>Find the volume of a sphere of radius [latex]R[\/latex] with a cap of height [latex]h[\/latex] removed from the top, as seen here.<\/p>\r\n\"This<\/span>\r\n[reveal-answer q=\"fs-id1167793638069\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793638069\"]\r\n\r\n[latex]\\frac{\\pi }{3}(h+R){(h-2R)}^{2}[\/latex] units3\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>","rendered":"

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1.\u00a0<\/strong>Derive the formula for the volume of a sphere using the slicing method.<\/p>\n<\/div>\n

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2.\u00a0<\/strong>Use the slicing method to derive the formula for the volume of a cone.<\/p>\n<\/div>\n<\/div>\n

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3.\u00a0<\/strong>Use the slicing method to derive the formula for the volume of a tetrahedron with side length [latex]a.[\/latex]<\/p>\n<\/div>\n<\/div>\n

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4.\u00a0<\/strong>Use the disk method to derive the formula for the volume of a trapezoidal cylinder.<\/p>\n<\/div>\n<\/div>\n

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5.\u00a0<\/strong>Explain when you would use the disk method versus the washer method. When are they interchangeable?<\/p>\n<\/div>\n<\/div>\n

For the following exercises (6-10), draw a typical slice and find the volume using the slicing method for the given volume.<\/p>\n

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6.\u00a0<\/strong>A pyramid with height 6 units and square base of side 2 units, as pictured here.<\/p>\n

\"This<\/span><\/p>\n

Show Solution<\/span><\/p>\n
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8 units3<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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7.\u00a0<\/strong>A pyramid with height 4 units and a rectangular base with length 2 units and width 3 units, as pictured here.<\/p>\n

\"This<\/span><\/p>\n<\/div>\n<\/div>\n

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8.\u00a0<\/strong>A tetrahedron with a base side of 4 units, as seen here.<\/p>\n

\"This<\/span><\/p>\n

Show Solution<\/span><\/p>\n
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[latex]\\frac{32}{3\\sqrt{2}}[\/latex] units3<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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9.\u00a0<\/strong>A pyramid with height 5 units, and an isosceles triangular base with lengths of 6 units and 8 units, as seen here.<\/p>\n

\"This<\/span><\/p>\n<\/div>\n<\/div>\n

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10.\u00a0<\/strong>A cone of radius [latex]r[\/latex] and height [latex]h[\/latex] has a smaller cone of radius [latex]r\\text{\/}2[\/latex] and height [latex]h\\text{\/}2[\/latex] removed from the top, as seen here. The resulting solid is called a frustum<\/em>.<\/p>\n

\"This<\/span><\/p>\n

Show Solution<\/span><\/p>\n
\n

[latex]\\frac{7\\pi }{12}h{r}^{2}[\/latex] units3<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises (11-16), draw an outline of the solid and find the volume using the slicing method.<\/p>\n

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11.\u00a0<\/strong>The base is a circle of radius [latex]a.[\/latex] The slices perpendicular to the base are squares.<\/p>\n<\/div>\n<\/div>\n

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12.\u00a0<\/strong>The base is a triangle with vertices [latex](0,0),(1,0),[\/latex] and [latex](0,1).[\/latex] Slices perpendicular to the xy<\/em>-plane are semicircles.<\/p>\n

Show Solution<\/span><\/p>\n
\n\"This<\/span>
\n[latex]\\frac{\\pi }{24}[\/latex] units3<\/sup><\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n

13.\u00a0<\/strong>The base is the region under the parabola [latex]y=1-{x}^{2}[\/latex] in the first quadrant. Slices perpendicular to the xy<\/em>-plane are squares.<\/p>\n<\/div>\n<\/div>\n

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14.\u00a0<\/strong>The base is the region under the parabola [latex]y=1-{x}^{2}[\/latex] and above the [latex]x\\text{-axis}\\text{.}[\/latex] Slices perpendicular to the [latex]y\\text{-axis}[\/latex] are squares.<\/p>\n

Show Solution<\/span><\/p>\n
\"This<\/span>
\n2 units3<\/sup><\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n

15.\u00a0<\/strong>The base is the region enclosed by [latex]y={x}^{2}[\/latex] and [latex]y=9.[\/latex] Slices perpendicular to the [latex]x[\/latex]-axis are right isosceles triangles.<\/p>\n<\/div>\n<\/div>\n

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16.\u00a0<\/strong>The base is the area between [latex]y=x[\/latex] and [latex]y={x}^{2}.[\/latex] Slices perpendicular to the [latex]x[\/latex]-axis are semicircles.<\/p>\n

Show Solution<\/span><\/p>\n
\n\"This<\/span>
\n[latex]\\frac{\\pi }{240}[\/latex] units3<\/sup><\/div>\n<\/div>\n<\/div>\n

For the following exercises (17-24), draw the region bounded by the curves. Then, use the disk method to find the volume when the region is rotated around the [latex]x[\/latex]-axis.<\/span><\/p>\n<\/div>\n

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17.\u00a0<\/strong>[latex]x+y=8,x=0,\\text{ and }y=0[\/latex]<\/p>\n<\/div>\n<\/div>\n

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18.\u00a0<\/strong>[latex]y=2{x}^{2},x=0,x=4,\\text{ and }y=0[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
\"This<\/span>
\n[latex]\\frac{4096\\pi }{5}[\/latex] units3<\/sup><\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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19.\u00a0<\/strong>[latex]y={e}^{x}+1,x=0,x=1,\\text{ and }y=0[\/latex]<\/p>\n<\/div>\n<\/div>\n

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20.\u00a0<\/strong>[latex]y={x}^{4},x=0,\\text{ and }y=1[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
\"This<\/span>
\n[latex]\\frac{8\\pi }{9}[\/latex] units3<\/sup><\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n

21.\u00a0<\/strong>[latex]y=\\sqrt{x},x=0,x=4,\\text{ and }y=0[\/latex]<\/p>\n<\/div>\n<\/div>\n

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22.\u00a0<\/strong>[latex]y= \\sin x,y= \\cos x,\\text{ and }x=0[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
\"This<\/span>
\n[latex]\\frac{\\pi }{2}[\/latex] units3<\/sup><\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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23.\u00a0<\/strong>[latex]y=\\frac{1}{x},x=2,\\text{ and }y=3[\/latex]<\/p>\n<\/div>\n<\/div>\n

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24.\u00a0<\/strong>[latex]{x}^{2}-{y}^{2}=9\\text{ and }x+y=9,y=0\\text{ and }x=0[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
\"This<\/span>
\n[latex]207\\pi[\/latex] units3<\/sup><\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises (25-32), draw the region bounded by the curves. Then, find the volume when the region is rotated around the [latex]y[\/latex]-axis.<\/p>\n

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25.\u00a0<\/strong>[latex]y=4-\\frac{1}{2}x,x=0,\\text{ and }y=0[\/latex]<\/p>\n<\/div>\n<\/div>\n

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26.\u00a0<\/strong>[latex]y=2{x}^{3},x=0,x=1,\\text{ and }y=0[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
\"This<\/span>
\n[latex]\\frac{4\\pi }{5}[\/latex] units3<\/sup><\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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27.\u00a0<\/strong>[latex]y=3{x}^{2},x=0,\\text{ and }y=3[\/latex]<\/p>\n<\/div>\n<\/div>\n

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28.\u00a0<\/strong>[latex]y=\\sqrt{4-{x}^{2}},y=0,\\text{ and }x=0[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
\"This<\/span>
\n[latex]\\frac{16\\pi }{3}[\/latex] units3<\/sup><\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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29.\u00a0<\/strong>[latex]y=\\frac{1}{\\sqrt{x+1}},x=0,\\text{ and }x=3[\/latex]<\/p>\n<\/div>\n<\/div>\n

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30.\u00a0<\/strong>[latex]x= \\sec (y)\\text{ and }y=\\frac{\\pi }{4},y=0\\text{ and }x=0[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
\"This<\/span>
\n[latex]\\pi[\/latex] units3<\/sup><\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n

31.\u00a0<\/strong>[latex]y=\\frac{1}{x+1},x=0,\\text{ and }x=2[\/latex]<\/p>\n<\/div>\n<\/div>\n

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32.\u00a0<\/strong>[latex]y=4-x,y=x,\\text{ and }x=0[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
\"This<\/span>
\n[latex]\\frac{16\\pi }{3}[\/latex] units3<\/sup><\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises (33-40), draw the region bounded by the curves. Then, find the volume when the region is rotated around the [latex]x[\/latex]-axis.<\/p>\n

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33.<\/strong> [latex]y=x+2,y=x+6,x=0,\\text{ and }x=5[\/latex]<\/p>\n<\/div>\n<\/div>\n

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34.\u00a0<\/strong>[latex]y={x}^{2}\\text{ and }y=x+2[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
\"This<\/span>
\n[latex]\\frac{72\\pi }{5}[\/latex] units3<\/sup><\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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35.\u00a0<\/strong>[latex]{x}^{2}={y}^{3}\\text{ and }{x}^{3}={y}^{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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36.\u00a0<\/strong>[latex]y=4-{x}^{2}\\text{ and }y=2-x[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
\"This<\/span>
\n[latex]\\frac{108\\pi }{5}[\/latex] units3<\/sup><\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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37. [T]<\/strong> [latex]y= \\cos x,y={e}^{\\text{\u2212}x},x=0,\\text{ and }x=1.2927[\/latex]<\/p>\n<\/div>\n<\/div>\n

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38.\u00a0<\/strong>[latex]y=\\sqrt{x}\\text{ and }y={x}^{2}[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
\"This<\/span>
\n[latex]\\frac{3\\pi }{10}[\/latex] units3<\/sup><\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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39.\u00a0<\/strong>[latex]y= \\sin x\\text{,}y=5 \\sin x,x=0\\text{ and }x=\\pi[\/latex]<\/p>\n<\/div>\n<\/div>\n

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40.\u00a0<\/strong>[latex]y=\\sqrt{1+{x}^{2}}\\text{ and }y=\\sqrt{4-{x}^{2}}[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
\"This<\/span>
\n[latex]2\\sqrt{6}\\pi[\/latex] units3<\/sup><\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises (41-45), draw the region bounded by the curves. Then, use the washer method to find the volume when the region is revolved around the [latex]y[\/latex]-axis.<\/p>\n

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41.\u00a0<\/strong>[latex]y=\\sqrt{x},x=4,\\text{ and }y=0[\/latex]<\/p>\n<\/div>\n<\/div>\n

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42.\u00a0<\/strong>[latex]y=x+2,y=2x-1,\\text{ and }x=0[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
\"This<\/span>
\n[latex]9\\pi[\/latex] units3<\/sup><\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n

43.\u00a0<\/strong>[latex]y=\\sqrt[3]{x}\\text{ and }y={x}^{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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44.\u00a0<\/strong>[latex]x={e}^{2y},x={y}^{2},y=0,\\text{ and }y=\\text{ln}(2)[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
\"This<\/span>
\n[latex]\\frac{\\pi }{20}(75-4{\\text{ln}}^{5}(2))[\/latex] units3<\/sup><\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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45.\u00a0<\/strong>[latex]x=\\sqrt{9-{y}^{2}},x={e}^{\\text{\u2212}y},y=0,\\text{ and }y=3[\/latex]<\/p>\n<\/div>\n<\/div>\n

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46.\u00a0<\/strong>Yogurt containers can be shaped like frustums. Rotate the line [latex]y=\\frac{1}{m}x[\/latex] around the [latex]y[\/latex]-axis to find the volume between [latex]y=a\\text{ and }y=b.[\/latex]<\/p>\n

\"This<\/span><\/p>\n

Show Solution<\/span><\/p>\n
\n

[latex]\\frac{{m}^{2}\\pi }{3}({b}^{3}-{a}^{3})[\/latex] units3<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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47.\u00a0<\/strong>Rotate the ellipse [latex]({x}^{2}\\text{\/}{a}^{2})+({y}^{2}\\text{\/}{b}^{2})=1[\/latex] around the [latex]x[\/latex]-axis to approximate the volume of a football, as seen here.<\/p>\n

\"This<\/span><\/p>\n<\/div>\n<\/div>\n

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48.\u00a0<\/strong>Rotate the ellipse [latex]({x}^{2}\\text{\/}{a}^{2})+({y}^{2}\\text{\/}{b}^{2})=1[\/latex] around the [latex]y[\/latex]-axis to approximate the volume of a football.<\/p>\n

Show Solution<\/span><\/p>\n
\n

[latex]\\frac{4{a}^{2}b\\pi }{3}[\/latex] units3<\/sup><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

\n
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49.\u00a0<\/strong>A better approximation of the volume of a football is given by the solid that comes from rotating [latex]y= \\sin x[\/latex] around the [latex]x[\/latex]-axis from [latex]x=0[\/latex] to [latex]x=\\pi .[\/latex] What is the volume of this football approximation, as seen here?<\/p>\n

\"This<\/span><\/p>\n<\/div>\n<\/div>\n

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50.\u00a0<\/strong>What is the volume of the Bundt cake that comes from rotating [latex]y= \\sin x[\/latex] around the [latex]y[\/latex]-axis from [latex]x=0[\/latex] to [latex]x=\\pi ?[\/latex]<\/p>\n

\"This<\/span><\/p>\n

Show Solution<\/span><\/p>\n
\n

[latex]2{\\pi }^{2}[\/latex] units3<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises (51-56), find the volume of the solid described.<\/p>\n

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51. <\/strong>The base is the region between [latex]y=x[\/latex] and [latex]y={x}^{2}.[\/latex] Slices perpendicular to the [latex]x[\/latex]-axis are semicircles.<\/p>\n<\/div>\n<\/div>\n

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52.\u00a0<\/strong>The base is the region enclosed by the generic ellipse [latex]({x}^{2}\\text{\/}{a}^{2})+({y}^{2}\\text{\/}{b}^{2})=1.[\/latex] Slices perpendicular to the [latex]x[\/latex]-axis are semicircles.<\/p>\n

Show Solution<\/span><\/p>\n
\n

[latex]\\frac{2a{b}^{2}\\pi }{3}[\/latex] units3<\/sup><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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53.\u00a0<\/strong>Bore a hole of radius [latex]a[\/latex] down the axis of a right cone and through the base of radius [latex]b,[\/latex] as seen here.<\/p>\n

\"This<\/span><\/p>\n<\/div>\n<\/div>\n

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54.\u00a0<\/strong>Find the volume common to two spheres of radius [latex]r[\/latex] with centers that are [latex]2h[\/latex] apart, as shown here.<\/p>\n

\"This<\/span><\/p>\n

Show Solution<\/span><\/p>\n
\n

[latex]\\frac{\\pi }{12}{(r+h)}^{2}(6r-h)[\/latex] units3<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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55.\u00a0<\/strong>Find the volume of a spherical cap of height [latex]h[\/latex] and radius [latex]r[\/latex] where [latex]h\n

\"This<\/span><\/p>\n<\/div>\n<\/div>\n

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56.\u00a0<\/strong>Find the volume of a sphere of radius [latex]R[\/latex] with a cap of height [latex]h[\/latex] removed from the top, as seen here.<\/p>\n

\"This<\/span><\/p>\n

Show Solution<\/span><\/p>\n
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[latex]\\frac{\\pi }{3}(h+R){(h-2R)}^{2}[\/latex] units3<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t

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Candela Citations<\/h3>\n\t\t\t\t\t
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