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

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.

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

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

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

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

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

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:

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:

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

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

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

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

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 =

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

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:

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

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

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

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