Page 321 - Calculus Demystified

308 Chapter 8 Solutions to Exercises
1. At position x in the base circle, the y-coordinate is √ 1 − x . Therefore the
2
half-disk slice has radius √ 1 − x and area π(1 − x )/2. The volume of
2
2
the solid is then
1 π(1 − x )
2
V = dx
−1 2
π x 3 1
= 2 x − 3
−1
π 1 −1
= 1 − − (−1) −
2 3 3
2π
= 3 .
2. We calculate the volume of half the solid, and then double the answer.
√
For 0 ≤ x ≤ 1/ 2, at position x in the base square, the y-coordinate
√
√
is 1/ 2 − x. Thus the disk slice has radius (1/ 2 − x) and area
√
π(1/ 2 − x) . Thus the volume of the solid is
2
TEAMFLY √
√

1/ 2
V = 2 π(1/ 2 − x) dx
2
0 √
! 2π 1 3 " 1/ 2
= − 3 √ − x
2
0
3
=− 2π 0 − √
1
3
3 2
π
= √ .
3 2
5
2 2
3. (a) π[x ] dx
2
3
(b) π[y ] dy
2 2
1
2 3/2 2
(c) 0 π[x + 1] dx
2
(d) π[5 − (x + 3)] dx
2
−1

Team-Fly
®