{"id":1123,"date":"2021-06-30T17:01:56","date_gmt":"2021-06-30T17:01:56","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/properties-of-the-definite-integral\/"},"modified":"2022-03-19T03:14:06","modified_gmt":"2022-03-19T03:14:06","slug":"properties-of-the-definite-integral","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/properties-of-the-definite-integral\/","title":{"raw":"Properties of the Definite Integral","rendered":"Properties of the Definite Integral"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Use geometry and the properties of definite integrals to evaluate them<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1170571711350\" class=\"bc-section section\">\r\n<div id=\"fs-id1170571711350\" class=\"bc-section section\">\r\n<p id=\"fs-id1170571711356\">The properties of indefinite integrals apply to definite integrals as well. Definite integrals also have properties that relate to the limits of integration. These properties, along with the rules of integration that we examine later in this chapter, help us manipulate expressions to evaluate definite integrals.<\/p>\r\n\r\n<div id=\"fs-id1170571711362\" class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">Properties of the Definite Integral<\/h3>\r\n\r\n<hr \/>\r\n\r\n<ol id=\"fs-id1170571711368\">\r\n \t<li>\r\n<div id=\"fs-id1170571711373\" class=\"equation\">[latex]\\displaystyle\\int_a^a f(x) dx = 0[\/latex]<\/div>\r\nIf the limits of integration are the same, the integral is just a line and contains no area.<\/li>\r\n \t<li>\r\n<div id=\"fs-id1170571812229\" class=\"equation\">[latex]\\displaystyle\\int_b^a f(x) dx = \u2212\\displaystyle\\int_a^b f(x) dx[\/latex]<\/div>\r\nIf the limits are reversed, then place a negative sign in front of the integral.<\/li>\r\n \t<li>\r\n<div id=\"fs-id1170571547568\" class=\"equation\">[latex]\\displaystyle\\int_a^b [f(x)+g(x)] dx = \\displaystyle\\int_a^b f(x) dx + \\displaystyle\\int_a^b g(x) dx[\/latex]<\/div>\r\nThe integral of a sum is the sum of the integrals.<\/li>\r\n \t<li>\r\n<div id=\"fs-id1170571807215\" class=\"equation\">[latex]\\displaystyle\\int_a^b [f(x)-g(x)] dx = \\displaystyle\\int_a^b f(x) dx - \\displaystyle\\int_a^b g(x) dx[\/latex]<\/div>\r\nThe integral of a difference is the difference of the integrals.<\/li>\r\n \t<li>\r\n<div id=\"fs-id1170572622483\" class=\"equation\">[latex]\\displaystyle\\int_a^b cf(x) dx= c \\displaystyle\\int_a^b f(x) dx[\/latex]<\/div>\r\nfor constant [latex]c[\/latex]. The integral of the product of a constant and a function is equal to the constant multiplied by the integral of the function.<\/li>\r\n \t<li>\r\n<div id=\"fs-id1170571775754\" class=\"equation\">[latex]\\displaystyle\\int_a^b f(x) dx = \\displaystyle\\int_a^c f(x) dx + \\displaystyle\\int_c^b f(x) dx[\/latex]<\/div>\r\nAlthough this formula normally applies when [latex]c[\/latex] is between [latex]a[\/latex] and [latex]b[\/latex], the formula holds for all values of [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex], provided [latex]f(x)[\/latex] is integrable on the largest interval.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"fs-id1170572274820\" class=\"textbox exercises\">\r\n<h3>Example: Using the Properties of the Definite Integral<\/h3>\r\nUse the properties of the definite integral to express the definite integral of [latex]f(x)=-3x^3+2x+2[\/latex] over the interval [latex][-2,1][\/latex] as the sum of three definite integrals.\r\n<div id=\"fs-id1170572274822\" class=\"exercise\">[reveal-answer q=\"fs-id1170571733908\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571733908\"]\r\n<p id=\"fs-id1170571733908\">Using integral notation, we have [latex]\\displaystyle\\int_{-2}^1 (-3x^3+2x+2) dx[\/latex]. We apply properties 3. and 5. to get<\/p>\r\n\r\n<div id=\"fs-id1170572558269\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{ll} \\displaystyle\\int_{-2}^1 (-3x^3+2x+2) dx &amp; = \\displaystyle\\int_{-2}^1 -3x^3 dx + \\displaystyle\\int_{-2}^1 2x dx + \\displaystyle\\int_{-2}^1 2 dx \\\\ &amp; =-3 \\displaystyle\\int_{-2}^1 x^3 dx + 2 \\displaystyle\\int_{-2}^1 x dx + \\displaystyle\\int_{-2}^1 2 dx \\end{array}[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572307259\" class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nUse the properties of the definite integral to express the definite integral of [latex]f(x)=6x^3-4x^2+2x-3[\/latex] over the interval [latex][1,3][\/latex] as the sum of four definite integrals.\r\n\r\n[reveal-answer q=\"134625\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"134625\"]Use the solving strategy from the previous example and the properties of definite integrals.[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1170572376231\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572376231\"]\r\n<p id=\"fs-id1170572376231\">[latex]6 \\displaystyle\\int_1^3 x^3 dx - 4 \\displaystyle\\int_1^3 x^2 dx + 2 \\displaystyle\\int_1^3 x dx - \\displaystyle\\int_1^3 3 dx[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572098842\" class=\"textbox exercises\">\r\n<h3>example: Using the Properties of the Definite Integral<\/h3>\r\n<p id=\"fs-id1170572098851\">If it is known that [latex]\\displaystyle\\int_0^8 f(x) dx = 10[\/latex] and [latex]\\displaystyle\\int_0^5 f(x) dx = 5[\/latex], find the value of [latex]\\displaystyle\\int_5^8 f(x) dx[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1170572344244\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572344244\"]\r\n<p id=\"fs-id1170572344244\">By property 6.,<\/p>\r\n\r\n<div id=\"fs-id1170572344247\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\int_a^b f(x) dx = \\displaystyle\\int_a^c f(x) dx + \\displaystyle\\int_c^b f(x) dx[\/latex].<\/div>\r\n<p id=\"fs-id1170572551964\">Thus,<\/p>\r\n\r\n<div id=\"fs-id1170572551967\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{lll} \\displaystyle\\int_0^8 f(x) dx &amp; = &amp; \\displaystyle\\int_0^5 f(x) dx + \\displaystyle\\int_5^8 f(x) dx \\\\ 10 &amp; = &amp; 5 + \\displaystyle\\int_5^8 f(x) dx \\\\ 5 &amp; = &amp; \\displaystyle\\int_5^8 f(x) dx \\end{array}[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\nWatch the following video to see the worked solution to Example: Using the Properties of the Definite Integral.\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/tto0E7yOSLo?controls=0&amp;start=1016&amp;end=1107&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>[reveal-answer q=\"266834\"]Closed Captioning and Transcript Information for Video[\/reveal-answer]\r\n[hidden-answer a=\"266834\"]For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.\r\n\r\nYou can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/5.2TheDefiniteIntegral1016to1107_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"5.2 The Definite Integral\" here (opens in new window)<\/a>.[\/hidden-answer]\r\n<div id=\"fs-id1170572500594\" class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1170572500602\">If it is known that [latex]\\displaystyle\\int_1^5 f(x) dx = -3[\/latex] and [latex]\\displaystyle\\int_2^5 f(x) dx = 4[\/latex], find the value of [latex]\\displaystyle\\int_1^2 f(x) dx[\/latex].<\/p>\r\n[reveal-answer q=\"583576\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"583576\"]Use the solving strategy from the last example and the rule on properties of definite integrals.[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1170572444364\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572444364\"]\r\n<p id=\"fs-id1170572444364\">\u22127<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"fs-id1170572444371\" class=\"bc-section section\">\r\n<h2>Comparison Properties of Integrals<\/h2>\r\n<p id=\"fs-id1170572444376\">A picture can sometimes tell us more about a function than the results of computations. Comparing functions by their graphs as well as by their algebraic expressions can often give new insight into the process of integration. Intuitively, we might say that if a function [latex]f(x)[\/latex] is above another function [latex]g(x)[\/latex], then the area between [latex]f(x)[\/latex] and the [latex]x[\/latex]-axis is greater than the area between [latex]g(x)[\/latex] and the [latex]x[\/latex]-axis. This is true depending on the interval over which the comparison is made. The properties of definite integrals are valid whether [latex]a&lt;b, \\, a=b[\/latex], or [latex]a&gt;b[\/latex]. The following properties, however, concern only the case [latex]a \\le b[\/latex], and are used when we want to compare the sizes of integrals.<\/p>\r\n\r\n<div id=\"fs-id1170571653089\" class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">Comparison Theorem<\/h3>\r\n\r\n<hr \/>\r\n\r\n<ol id=\"fs-id1170572293423\">\r\n \t<li>If [latex]f(x) \\ge 0[\/latex] for [latex]a \\le x \\le b[\/latex], then\r\n<div id=\"fs-id1170570996273\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\int_a^b f(x) dx \\ge 0[\/latex].<\/div><\/li>\r\n \t<li>If [latex]f(x) \\ge g(x)[\/latex] for [latex]a \\le x \\le b[\/latex], then\r\n<div id=\"fs-id1170573750186\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\int_a^b f(x) dx \\ge \\displaystyle\\int_a^b g(x) dx[\/latex].<\/div><\/li>\r\n \t<li>If [latex]m[\/latex] and [latex]M[\/latex] are constants such that [latex]m \\le f(x) \\le M[\/latex] for [latex]a \\le x \\le b[\/latex], then\r\n<div id=\"fs-id1170573255165\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]m(b-a) \\le \\displaystyle\\int_a^b f(x) dx \\le M(b-a)[\/latex].<\/div><\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"fs-id1170571769594\" class=\"textbox exercises\">\r\n<h3>example: Comparing Two Functions over a Given Interval<\/h3>\r\nCompare [latex]f(x)=\\sqrt{1+x^2}[\/latex] and [latex]g(x)=\\sqrt{1+x}[\/latex] over the interval [latex][0,1][\/latex].\r\n[reveal-answer q=\"fs-id1170572628469\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572628469\"]\r\n<p id=\"fs-id1170572628469\">Graphing these functions is necessary to understand how they compare over the interval [latex][0,1][\/latex]. Initially, when graphed on a graphing calculator, [latex]f(x)[\/latex] appears to be above [latex]g(x)[\/latex] everywhere. However, on the interval [latex][0,1][\/latex], the graphs appear to be on top of each other. We need to zoom in to see that, on the interval [latex][0,1], \\, g(x)[\/latex] is above [latex]f(x)[\/latex]. The two functions intersect at [latex]x=0[\/latex] and [latex]x=1[\/latex] (Figure 9).<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"560\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11204030\/CNX_Calc_Figure_05_02_011.jpg\" alt=\"A graph of the function f(x) = sqrt(1 + x^2) in red and g(x) = sqrt(1 + x) in blue over [-2, 3]. The function f(x) appears above g(x) except over the interval [0,1]. A second, zoomed-in graph shows this interval more clearly.\" width=\"560\" height=\"241\" \/> Figure 9. (a) The function [latex]f(x)[\/latex] appears above the function [latex]g(x)[\/latex] except over the interval [latex][0,1][\/latex] (b) Viewing the same graph with a greater zoom shows this more clearly.[\/caption]\r\n<p id=\"fs-id1170572331851\">We can see from the graph that over the interval [latex][0,1], \\, g(x) \\ge f(x)[\/latex]. Comparing the integrals over the specified interval [latex][0,1][\/latex], we also see that [latex]\\displaystyle\\int_0^1 g(x) dx \\ge \\displaystyle\\int_0^1 f(x) dx[\/latex] (Figure 10). The thin, red-shaded area shows just how much difference there is between these two integrals over the interval [latex][0,1][\/latex].<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"560\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11204034\/CNX_Calc_Figure_05_02_012.jpg\" alt=\"A graph showing the functions f(x) = sqrt(1 + x^2) and g(x) = sqrt(1 + x) over [-3, 3]. The area under g(x) in quadrant one over [0,1] is shaded. The area under g(x) and f(x) is included in this shaded area. The second, zoomed-in graph shows more clearly that equality between the functions only holds at the endpoints.\" width=\"560\" height=\"241\" \/> Figure 10. (a) The graph shows that over the interval [latex][0,1], \\, g(x) \\ge f(x)[\/latex], where equality holds only at the endpoints of the interval. (b) Viewing the same graph with a greater zoom shows this more clearly.[\/caption][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\nWatch the following video to see the worked solution to Example: Comparing Two Functions over a Given Interval.\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/tto0E7yOSLo?controls=0&amp;start=1240&amp;end=1411&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>[reveal-answer q=\"266833\"]Closed Captioning and Transcript Information for Video[\/reveal-answer]\r\n[hidden-answer a=\"266833\"]For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.\r\n\r\nYou can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/5.2TheDefiniteIntegral1240to1411_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"5.2 The Definite Integral\" here (opens in new window)<\/a>.[\/hidden-answer]\r\n<div id=\"1622857412.000500\" class=\"c-virtual_list__item\" tabindex=\"-1\" role=\"listitem\" aria-expanded=\"false\" data-qa=\"virtual-list-item\">\r\n<div class=\"c-message_kit__background p-message_pane_message__message c-message_kit__message\" role=\"document\" data-qa=\"message_container\" data-qa-unprocessed=\"false\" data-qa-placeholder=\"false\">\r\n<div class=\"c-message_kit__hover\" role=\"document\" data-qa-hover=\"true\">\r\n<div class=\"c-message_kit__actions c-message_kit__actions--above\">\r\n<div class=\"c-message_kit__gutter\">\r\n<div class=\"c-message_kit__gutter__right\" data-qa=\"message_content\">\r\n<div class=\"c-message_kit__blocks c-message_kit__blocks--rich_text\">\r\n<div class=\"c-message__message_blocks c-message__message_blocks--rich_text\">\r\n<div class=\"p-block_kit_renderer\" data-qa=\"block-kit-renderer\">\r\n<div class=\"p-block_kit_renderer__block_wrapper p-block_kit_renderer__block_wrapper--first\">\r\n<div class=\"p-rich_text_block\" dir=\"auto\"><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Use geometry and the properties of definite integrals to evaluate them<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1170571711350\" class=\"bc-section section\">\n<div id=\"fs-id1170571711350\" class=\"bc-section section\">\n<p id=\"fs-id1170571711356\">The properties of indefinite integrals apply to definite integrals as well. Definite integrals also have properties that relate to the limits of integration. These properties, along with the rules of integration that we examine later in this chapter, help us manipulate expressions to evaluate definite integrals.<\/p>\n<div id=\"fs-id1170571711362\" class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">Properties of the Definite Integral<\/h3>\n<hr \/>\n<ol id=\"fs-id1170571711368\">\n<li>\n<div id=\"fs-id1170571711373\" class=\"equation\">[latex]\\displaystyle\\int_a^a f(x) dx = 0[\/latex]<\/div>\n<p>If the limits of integration are the same, the integral is just a line and contains no area.<\/li>\n<li>\n<div id=\"fs-id1170571812229\" class=\"equation\">[latex]\\displaystyle\\int_b^a f(x) dx = \u2212\\displaystyle\\int_a^b f(x) dx[\/latex]<\/div>\n<p>If the limits are reversed, then place a negative sign in front of the integral.<\/li>\n<li>\n<div id=\"fs-id1170571547568\" class=\"equation\">[latex]\\displaystyle\\int_a^b [f(x)+g(x)] dx = \\displaystyle\\int_a^b f(x) dx + \\displaystyle\\int_a^b g(x) dx[\/latex]<\/div>\n<p>The integral of a sum is the sum of the integrals.<\/li>\n<li>\n<div id=\"fs-id1170571807215\" class=\"equation\">[latex]\\displaystyle\\int_a^b [f(x)-g(x)] dx = \\displaystyle\\int_a^b f(x) dx - \\displaystyle\\int_a^b g(x) dx[\/latex]<\/div>\n<p>The integral of a difference is the difference of the integrals.<\/li>\n<li>\n<div id=\"fs-id1170572622483\" class=\"equation\">[latex]\\displaystyle\\int_a^b cf(x) dx= c \\displaystyle\\int_a^b f(x) dx[\/latex]<\/div>\n<p>for constant [latex]c[\/latex]. The integral of the product of a constant and a function is equal to the constant multiplied by the integral of the function.<\/li>\n<li>\n<div id=\"fs-id1170571775754\" class=\"equation\">[latex]\\displaystyle\\int_a^b f(x) dx = \\displaystyle\\int_a^c f(x) dx + \\displaystyle\\int_c^b f(x) dx[\/latex]<\/div>\n<p>Although this formula normally applies when [latex]c[\/latex] is between [latex]a[\/latex] and [latex]b[\/latex], the formula holds for all values of [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex], provided [latex]f(x)[\/latex] is integrable on the largest interval.<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1170572274820\" class=\"textbox exercises\">\n<h3>Example: Using the Properties of the Definite Integral<\/h3>\n<p>Use the properties of the definite integral to express the definite integral of [latex]f(x)=-3x^3+2x+2[\/latex] over the interval [latex][-2,1][\/latex] as the sum of three definite integrals.<\/p>\n<div id=\"fs-id1170572274822\" class=\"exercise\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170571733908\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170571733908\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571733908\">Using integral notation, we have [latex]\\displaystyle\\int_{-2}^1 (-3x^3+2x+2) dx[\/latex]. We apply properties 3. and 5. to get<\/p>\n<div id=\"fs-id1170572558269\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{ll} \\displaystyle\\int_{-2}^1 (-3x^3+2x+2) dx & = \\displaystyle\\int_{-2}^1 -3x^3 dx + \\displaystyle\\int_{-2}^1 2x dx + \\displaystyle\\int_{-2}^1 2 dx \\\\ & =-3 \\displaystyle\\int_{-2}^1 x^3 dx + 2 \\displaystyle\\int_{-2}^1 x dx + \\displaystyle\\int_{-2}^1 2 dx \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572307259\" class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Use the properties of the definite integral to express the definite integral of [latex]f(x)=6x^3-4x^2+2x-3[\/latex] over the interval [latex][1,3][\/latex] as the sum of four definite integrals.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q134625\">Hint<\/span><\/p>\n<div id=\"q134625\" class=\"hidden-answer\" style=\"display: none\">Use the solving strategy from the previous example and the properties of definite integrals.<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572376231\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572376231\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572376231\">[latex]6 \\displaystyle\\int_1^3 x^3 dx - 4 \\displaystyle\\int_1^3 x^2 dx + 2 \\displaystyle\\int_1^3 x dx - \\displaystyle\\int_1^3 3 dx[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572098842\" class=\"textbox exercises\">\n<h3>example: Using the Properties of the Definite Integral<\/h3>\n<p id=\"fs-id1170572098851\">If it is known that [latex]\\displaystyle\\int_0^8 f(x) dx = 10[\/latex] and [latex]\\displaystyle\\int_0^5 f(x) dx = 5[\/latex], find the value of [latex]\\displaystyle\\int_5^8 f(x) dx[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572344244\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572344244\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572344244\">By property 6.,<\/p>\n<div id=\"fs-id1170572344247\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\int_a^b f(x) dx = \\displaystyle\\int_a^c f(x) dx + \\displaystyle\\int_c^b f(x) dx[\/latex].<\/div>\n<p id=\"fs-id1170572551964\">Thus,<\/p>\n<div id=\"fs-id1170572551967\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{lll} \\displaystyle\\int_0^8 f(x) dx & = & \\displaystyle\\int_0^5 f(x) dx + \\displaystyle\\int_5^8 f(x) dx \\\\ 10 & = & 5 + \\displaystyle\\int_5^8 f(x) dx \\\\ 5 & = & \\displaystyle\\int_5^8 f(x) dx \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to Example: Using the Properties of the Definite Integral.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/tto0E7yOSLo?controls=0&amp;start=1016&amp;end=1107&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q266834\">Closed Captioning and Transcript Information for Video<\/span><\/p>\n<div id=\"q266834\" class=\"hidden-answer\" style=\"display: none\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/5.2TheDefiniteIntegral1016to1107_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;5.2 The Definite Integral&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<div id=\"fs-id1170572500594\" class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1170572500602\">If it is known that [latex]\\displaystyle\\int_1^5 f(x) dx = -3[\/latex] and [latex]\\displaystyle\\int_2^5 f(x) dx = 4[\/latex], find the value of [latex]\\displaystyle\\int_1^2 f(x) dx[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q583576\">Hint<\/span><\/p>\n<div id=\"q583576\" class=\"hidden-answer\" style=\"display: none\">Use the solving strategy from the last example and the rule on properties of definite integrals.<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572444364\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572444364\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572444364\">\u22127<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572444371\" class=\"bc-section section\">\n<h2>Comparison Properties of Integrals<\/h2>\n<p id=\"fs-id1170572444376\">A picture can sometimes tell us more about a function than the results of computations. Comparing functions by their graphs as well as by their algebraic expressions can often give new insight into the process of integration. Intuitively, we might say that if a function [latex]f(x)[\/latex] is above another function [latex]g(x)[\/latex], then the area between [latex]f(x)[\/latex] and the [latex]x[\/latex]-axis is greater than the area between [latex]g(x)[\/latex] and the [latex]x[\/latex]-axis. This is true depending on the interval over which the comparison is made. The properties of definite integrals are valid whether [latex]a<b, \\, a=b[\/latex], or [latex]a>b[\/latex]. The following properties, however, concern only the case [latex]a \\le b[\/latex], and are used when we want to compare the sizes of integrals.<\/p>\n<div id=\"fs-id1170571653089\" class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">Comparison Theorem<\/h3>\n<hr \/>\n<ol id=\"fs-id1170572293423\">\n<li>If [latex]f(x) \\ge 0[\/latex] for [latex]a \\le x \\le b[\/latex], then\n<div id=\"fs-id1170570996273\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\int_a^b f(x) dx \\ge 0[\/latex].<\/div>\n<\/li>\n<li>If [latex]f(x) \\ge g(x)[\/latex] for [latex]a \\le x \\le b[\/latex], then\n<div id=\"fs-id1170573750186\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\int_a^b f(x) dx \\ge \\displaystyle\\int_a^b g(x) dx[\/latex].<\/div>\n<\/li>\n<li>If [latex]m[\/latex] and [latex]M[\/latex] are constants such that [latex]m \\le f(x) \\le M[\/latex] for [latex]a \\le x \\le b[\/latex], then\n<div id=\"fs-id1170573255165\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]m(b-a) \\le \\displaystyle\\int_a^b f(x) dx \\le M(b-a)[\/latex].<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1170571769594\" class=\"textbox exercises\">\n<h3>example: Comparing Two Functions over a Given Interval<\/h3>\n<p>Compare [latex]f(x)=\\sqrt{1+x^2}[\/latex] and [latex]g(x)=\\sqrt{1+x}[\/latex] over the interval [latex][0,1][\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572628469\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572628469\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572628469\">Graphing these functions is necessary to understand how they compare over the interval [latex][0,1][\/latex]. Initially, when graphed on a graphing calculator, [latex]f(x)[\/latex] appears to be above [latex]g(x)[\/latex] everywhere. However, on the interval [latex][0,1][\/latex], the graphs appear to be on top of each other. We need to zoom in to see that, on the interval [latex][0,1], \\, g(x)[\/latex] is above [latex]f(x)[\/latex]. The two functions intersect at [latex]x=0[\/latex] and [latex]x=1[\/latex] (Figure 9).<\/p>\n<div style=\"width: 570px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11204030\/CNX_Calc_Figure_05_02_011.jpg\" alt=\"A graph of the function f(x) = sqrt(1 + x^2) in red and g(x) = sqrt(1 + x) in blue over &#091;-2, 3&#093;. The function f(x) appears above g(x) except over the interval &#091;0,1&#093;. A second, zoomed-in graph shows this interval more clearly.\" width=\"560\" height=\"241\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 9. (a) The function [latex]f(x)[\/latex] appears above the function [latex]g(x)[\/latex] except over the interval [latex][0,1][\/latex] (b) Viewing the same graph with a greater zoom shows this more clearly.<\/p>\n<\/div>\n<p id=\"fs-id1170572331851\">We can see from the graph that over the interval [latex][0,1], \\, g(x) \\ge f(x)[\/latex]. Comparing the integrals over the specified interval [latex][0,1][\/latex], we also see that [latex]\\displaystyle\\int_0^1 g(x) dx \\ge \\displaystyle\\int_0^1 f(x) dx[\/latex] (Figure 10). The thin, red-shaded area shows just how much difference there is between these two integrals over the interval [latex][0,1][\/latex].<\/p>\n<div style=\"width: 570px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11204034\/CNX_Calc_Figure_05_02_012.jpg\" alt=\"A graph showing the functions f(x) = sqrt(1 + x^2) and g(x) = sqrt(1 + x) over &#091;-3, 3&#093;. The area under g(x) in quadrant one over &#091;0,1&#093; is shaded. The area under g(x) and f(x) is included in this shaded area. The second, zoomed-in graph shows more clearly that equality between the functions only holds at the endpoints.\" width=\"560\" height=\"241\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 10. (a) The graph shows that over the interval [latex][0,1], \\, g(x) \\ge f(x)[\/latex], where equality holds only at the endpoints of the interval. (b) Viewing the same graph with a greater zoom shows this more clearly.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to Example: Comparing Two Functions over a Given Interval.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/tto0E7yOSLo?controls=0&amp;start=1240&amp;end=1411&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q266833\">Closed Captioning and Transcript Information for Video<\/span><\/p>\n<div id=\"q266833\" class=\"hidden-answer\" style=\"display: none\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/5.2TheDefiniteIntegral1240to1411_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;5.2 The Definite Integral&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<div id=\"1622857412.000500\" class=\"c-virtual_list__item\" tabindex=\"-1\" role=\"listitem\" aria-expanded=\"false\" data-qa=\"virtual-list-item\">\n<div class=\"c-message_kit__background p-message_pane_message__message c-message_kit__message\" role=\"document\" data-qa=\"message_container\" data-qa-unprocessed=\"false\" data-qa-placeholder=\"false\">\n<div class=\"c-message_kit__hover\" role=\"document\" data-qa-hover=\"true\">\n<div class=\"c-message_kit__actions c-message_kit__actions--above\">\n<div class=\"c-message_kit__gutter\">\n<div class=\"c-message_kit__gutter__right\" data-qa=\"message_content\">\n<div class=\"c-message_kit__blocks c-message_kit__blocks--rich_text\">\n<div class=\"c-message__message_blocks c-message__message_blocks--rich_text\">\n<div class=\"p-block_kit_renderer\" data-qa=\"block-kit-renderer\">\n<div class=\"p-block_kit_renderer__block_wrapper p-block_kit_renderer__block_wrapper--first\">\n<div class=\"p-rich_text_block\" dir=\"auto\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1123\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>5.2 The Definite Integral. <strong>Authored by<\/strong>: Ryan Melton. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 2. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction\">https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t 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