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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.
Section 7.1 Measurement: Area and Curve Length
Area Idea
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.
Example 4
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.
Integrating in x
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:
Integrating in y
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:
Curve Length Idea
We can use line segments for an approximation.
• “Cut” [a, b] into n subintervals each of width
• Form the polygonal arc Cn made from the n line
segments joining the consecutive partition points.
• Add the length of each segment to get the length of Cn.
• Length of the curve =
Theorem
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: