Page 244 - Calculus Demystified

CHAPTER 8
Applications of the Integral
231
243 2 · 243 0 0
= 2π · − − −
2 5 2 5
243
= 2π ·
10
243π
= .
5
You Try It: Use the method of cylindrical shells to calculate the volume enclosed
when the region 0 ≤ y ≤ sin x,0 ≤ x ≤ π/2, is rotated about the y-axis.

8.2.3 DIFFERENT AXES
Sometimes it is convenient to rotate a curve about some line other than the
coordinate axes. We now provide a couple of examples of that type of problem.

EXAMPLE 8.11
Use the method of washers to calculate the volume of the solid enclosed
√
when the curve y = x,1 ≤ x ≤ 4, isrotated about the line y =−1. See
Fig. 8.21.

y

y = √x

x

Fig. 8.21

SOLUTION
The key is to notice that, at position x, the segment to be rotated has height
√ √
x − (−1)—the distance from the point (x, x) on the curve to the line
√
2
y =−1. Thus the disk generated has area A(x) = π ·( x +1) . The resulting