{"id":277,"date":"2021-02-04T00:46:02","date_gmt":"2021-02-04T00:46:02","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&p=277"},"modified":"2024-04-16T17:28:53","modified_gmt":"2024-04-16T17:28:53","slug":"the-area-problem-and-integral-calculus","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/the-area-problem-and-integral-calculus\/","title":{"raw":"The Area Problem and Integral Calculus","rendered":"The Area Problem and Integral Calculus"},"content":{"raw":"
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Learning Outcomes<\/h3>\r\n
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  • Describe the area problem and how it was solved by the integral<\/span><\/li>\r\n \t
  • Explain how the idea of a limit is involved in solving the area problem<\/span><\/li>\r\n \t
  • Recognize the total cost as the area under the marginal cost curve<\/li>\r\n<\/ul>\r\n<\/div>\r\n

    We now turn our attention to a classic question from calculus. Many quantities in physics\u2014for example, quantities of work\u2014may be interpreted as the area under a curve. This leads us to ask the question: How can we find the area between the graph of a function and the [latex]x[\/latex]-axis over an interval (Figure 7)?<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"325\"]\"A Figure 7. The Area Problem: How do we find the area of the shaded region?[\/caption]\r\n

    As in the answer to our previous questions on velocity, we first try to approximate the solution. We approximate the area by dividing up the interval [latex][a,b][\/latex] into smaller intervals in the shape of rectangles. The approximation of the area comes from adding up the areas of these rectangles (Figure 8).\u00a0Recall that the area of a rectangle can be found simply by taking the length times the width.<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"325\"]\"The Figure 8. The area of the region under the curve is approximated by summing the areas of thin rectangles.[\/caption]\r\n

    As the widths of the rectangles become smaller (approach zero), the sums of the areas of the rectangles approach the area between the graph of [latex]f(x)[\/latex] and the [latex]x[\/latex]-axis over the interval [latex][a,b][\/latex]. Once again, we find ourselves taking a limit. Limits of this type serve as a basis for the definition of the definite integral. Integral calculus<\/strong> is the study of integrals and their applications.<\/p>\r\n\r\n

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    Example: Estimation Using Rectangles<\/h3>\r\n

    Estimate the area between the [latex]x[\/latex]-axis and the graph of [latex]f(x)=x^2+1[\/latex] over the interval [latex][0,3][\/latex] by using the three rectangles shown in Figure 9.<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"325\"]\"A Figure 9. The area of the region under the curve of [latex]f(x)=x^2+1[\/latex] can be estimated using rectangles.[\/caption]\r\n

    <\/div>\r\n[reveal-answer q=\"fs-id1170573398981\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170573398981\"]The areas of the three rectangles are 1 unit2<\/sup>, 2 unit2<\/sup>, and 5 unit2<\/sup>. Using these rectangles, our area estimate is 8 unit2<\/sup>.[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n\r\n[caption]Watch the following video to see the worked solution to Example: Estimation Using Rectangles[\/caption]\r\n\r\n