Page 273 - Calculus Demystified

CHAPTER 8
260
Exercises Applications of the Integral
1. A solid has base the unit circle and vertical slices, parallel to the y-axis,
which are half-disks. Calculate the volume of this solid.
2. A solid has base a unit square with center at the origin and vertices on the
x- and y-axes. The vertical cross-section of this solid, parallel to the y-axis,
is a disk. What is the volume of this solid?
3. Set up the integral to calculate the volume enclosed when the indicated
curve over the indicated interval is rotated about the indicated line. Do not
evaluate the integral.

(a) y = x 2 2 ≤ x ≤ 5 x-axis
√
(b) y = x 1 ≤ x ≤ 9 y-axis
(c) y = x 3/2 0 ≤ x ≤ 2 y =−1
(d) y = x + 3 −1 ≤ x ≤ 2 y = 5
(e) y = x 1/2 4 ≤ x ≤ 6 x =−2
(f) y = sin x 0 ≤ x ≤ π/2 y = 0

4. Set up the integral to evaluate the indicated surface area. Do not evaluate.
(a) The area of the surface obtained when y = x 2/3 ,0 ≤ x ≤ 4, is
rotated about the x-axis.
(b) The area of the surface obtained when y = x 1/2 ,0 ≤ x ≤ 3, is
rotated about the y-axis.
2
(c) The area of the surface obtained when y = x ,0 ≤ x ≤ 3, is rotated
about the line y =−2.
(d) The area of the surface obtained when y = sin x,0 ≤ x ≤ π,is
rotated about the x-axis.
(e) The area of the surface obtained when y = x 1/2 ,1 ≤ x ≤ 4, is
rotated about the line x =−2.
3
(f) The area of the surface obtained when y = x ,0 ≤ x ≤ 1, is rotated
about the x-axis.
5. A water tank has a submerged window that is in the shape of a circle of
radius 2 feet. The center of this circular window is 8 feet below the surface.
Set up, but do not calculate, the integral for the pressure on the lower half
of this window—assuming that water weighs 62.4 pounds per cubic foot.
6. Aswimming pool isV-shaped. Each end of the pool is an inverted equilateral
triangle of side 10 feet. The pool is 25 feet long. The pool is full. Set up,
but do not calculate, the integral for the pressure on one end of the pool.
7. A man climbs a ladder with a 100 pound sack of sand that is leaking one
pound per minute. If he climbs steadily at the rate of 5 feet per minute, and