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Suppose you are estimating the area bounded by some curve and the $x$ -axis, between the lines $x=3$ and $x=5 .$ Determine the width of each rectangle if the number of rectangles to be used in the approximation is (a) $4,$ (b) 10 ,(c) 100.

(a) 0.5(b) 0.2(c) 0.02

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 4

Approximation of Areas

Integrals

Campbell University

Baylor University

University of Michigan - Ann Arbor

Lectures

05:53

In mathematics, an indefinite integral is an integral whose integrand is not known in terms of elementary functions. An indefinite integral is usually encountered when integrating functions that are not elementary functions themselves.

40:35

In mathematics, integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function of a real variable (often called "the integrand"), an antiderivative is a function whose derivative is the given function. The area under a real-valued function of a real variable is the integral of the function, provided it is defined on a closed interval around a given point. It is a basic result of calculus that an antiderivative always exists, and is equal to the original function evaluated at the upper limit of integration.

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(a) Use the rectangles in …

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in problem mine. We won't use the rectangular in these two graphs to approximate the area bounded by the region. This curve, which is why equals five divided by X and X, equals one here and calls five years on the X axis For this graph. By calculating submission off the area of these rectangular, we can say that the area equals one multiplied by the height. She's five plus one want to blow it by the right, which is almost 2.5 plus one. It's applied by the height, which is almost 1.5 plus one multiplied by the height, which is about one point to equals. Five. Multiply five plus four. She's nine. It equals approximately 1.2 for this graph. We can do the same here by getting this a new approximation for this area. Volume a submission off direct anglers. Each rectangular has a with off 0.5 and it has a different type. The first night it's five plus. The second right is about three 0.2 plus third height is 2.5 plus we have the fourth height is about to Plus we have here. The height is about 1.5. Then we have about 1.3. Then we have about 1.2. Then we have about 1.51 point one equals, huh? Multiplied by five plus 3.2 plus 2.5 plus two plus 1.5 plus 1.3 plus 1.2 plus 1.1 equals 8.9 approximately equals a point. Mind for both. Be off the problem. We want to continue this process to obtain a more accurate approximation. We can see that we have here used more Richt anglers, which give more accurate than they left the graph. We can continue by increasing the number of rectangular that approximates the that approximates in the area. It starts at one and at five. 123 Food we can use, for example, a rectangular off. 0, 00.25 Then we will have here one toe. 34 four decked anglers in each one with 1234 Yeah, the year 12 34 we have here. One, 234 Then direct Anglers starts at every line till the end of the second life starts. At the first line, we can see that the error is getting too small. This is an error we started at the line. I exist. I'm We're calculating the area of these rectangular these rectangles. It's about what four multiplied by 4 16 nicked anglers. We will get more accurate results.

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