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.

Area Idea

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

Example 1

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

Example 2

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

Example 3

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

Example 4

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

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:

Example 4

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

Example 5

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

Example 6

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

Curve Length Idea

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

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 =

Curve Length Idea

Below is shown a picture for C4 for the function graphed from x = 0 to x = 2.

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:

Example 7

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

Example 8

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

Example 9

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