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Published byPhyllis Charla Griffith Modified over 4 years ago

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In this section, we will investigate the process for finding the area between two curves and also the length of a given curve.

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We already have established is the signed area of the region between the curve y = f(x) and the x-axis. What if we wanted to find the area between two curves, y = f(x) and y = g(x) from x = a to x = b.

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We find this area (not signed area) by calculating:

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Find the area of the region bounded by the sine and cosine curves between and.

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Find the area of the region bounded by the curves and.

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Find the area of the region bounded by the curves,, and.

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Problem: There is not a distinct “top” and “bottom” curve, but there is a distinct “right” and “left” curve.

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Let R be the region bounded above by y = f(x), bounded below by y = g(x), on the left by x = a, and on the right by x = b. The area of R is:

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Let R be the region bounded on the right by the curve x = f(y), bounded on the left by x = g(y), on the bottom by y = c, and on the top by y = d. The area of R is:

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Find the area of the region bounded by the curves,, and.

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Find the area of the region bounded by the curves and using both techniques.

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Find the area of the region bounded by the curves,, and using both techniques.

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How long is the curve y = f(x) from x = a to x = b?

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We can use line segments for an approximation. “Cut” [a, b] into n subintervals each of width Form the polygonal arc C n made from the n line segments joining the consecutive partition points. Add the length of each segment to get the length of C n. Length of the curve =

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Below is shown a picture for C 4 for the function graphed from x = 0 to x = 2.

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Suppose f is differentiable on [a, b]. Then the length of the curve y = f(x) from x = a to x = b is given by:

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Find the length of the curve from x = 1 to x = 3.

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Find the length of the curve from x = 0 to x = 1.5.

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Estimate the length of the curve from x = 0 to x = 2 using 50 midpoint rectangles.

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