CHAPTER 21
Area and Arc Length
21.1
Sketch and find the area of the region to the left of the parabola x = 2y
2 , to the right of the y-axis, and between
y — 1 and y — 3.
See Fig. 21-1. The base of the region is the y-axis. The area is given by the integral
Fig. 21-1
Fig. 21-2
21.2
Sketch and find the area of the region above the line y = 3x - 2, in the first quadrant, and below the line
y = 4.
See Fig. 21-2, The region has a base on the y-axis. We must solve y = 3x - 2 for x:
Then the area is
21.3
Sketch and find the area of the region between the curve y = x
3
and the lines y = — x and y = 1.
See Fig. 21-3. The lower boundary of the region is y=—x and the upper boundary is y = x
3 . Hence,
the area is given by the integral
In Problems 21.4-21.16, sketch the indicated region and find its area.
21.4
Fig. 21-3
Fig. 21-4
163
The bounded region between the curves y = x
2
and y = x
3 .
See Fig. 21-4. The curves intersect at (0,0) and(l, 1). Between x = 0 and x = 1, y = x
2
lies above
y = x
3 . The area of the region between them is