in this section, we will investigate the process for finding the area between two curves and also...
TRANSCRIPT
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: