Page 245 - Calculus Demystified

CHAPTER 8
232
aggregate volume is Applications of the Integral
4
√ 2
V = π · x + 1 dx
1

4 √
= π x + 2 x + 1 dx
1
2 3/2 4
x 2x
= π + + x
2 3/2 1
2 2
4 2 · 8 1 2 · 1
= π · + + 4 − π · + + 1
2 3/2 2 3/2
119
= π.
6
√
You Try It: Calculate the volume inside the surface generated when y = 3 x +x
is rotated about the line y =−3, 1 ≤ x ≤ 4.
EXAMPLE 8.12
Calculate the volume of the solid enclosed when the area between the
2
2
curves x = (y − 2) + 1 and x =−(y − 2) + 9 isrotated about the line
y =−2.

y

_ 2 _ _ 2
x = (y 2) + 1 x = (y 2) + 9

x

Fig. 8.22

SOLUTION
Solving the equations simultaneously, we find that the points of intersec-
tion are (5, 0) and (5, 4). The region between the two curves is illustrated in
Fig. 8.22.
At height y, the horizontal segment that is to be rotated stretches from
2
2
((y − 2) + 1,y) to (−(y − 2) + 9,y). Thus the cylindrical shell that is