By
Steven G. Krantz — McGrawHill Professional
Updated on Aug 31, 2011
Review the following concepts if needed:
Applications of the Integral Practice Test

A solid has base the unit circle and vertical slices, parallel to the y axis, which are halfdisks. Calculate the volume of this solid.

A solid has base a unit square with center at the origin and vertices on the x  and y axes. The vertical crosssection of this solid, parallel to the y axis, is a disk. What is the volume of this solid?

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.

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.
(c) The area of the surface obtained when y = x ^{2} , 0 ≤ x ≤ 3, is rotated about the line y = −2.
(d) The area of the surface obtained when y = sin x, 0 ≤ x ≤ , 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.
(f) The area of the surface obtained when y = x ^{3} , 0 ≤ x ≤ 1, is rotated about the x axis.

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.

A swimming pool is Vshaped. 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.

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 if the ladder is 40 feet high, then how much work does he do in climbing the ladder?

Because of a prevailing wind, the force that opposes a certain runner is 3 x ^{2} + 4 x + 6 pounds at position x. How much work does this runner perform as he runs from x = 3 to x = 100 (with distance measured in feet)?

Set up, but do not evaluate, the integrals for each of the following arc length problems.
(a) The length of the curve y = sin x , 0 ≤ x ≤
(b) The length of the curve x ^{2} = y ^{3}, 1 ≤ x ≤ 8
(c) The length of the curve cos y = x , 0 ≤ y ≤ π/2
(d) The length of the curve y = x ^{2}, 1 ≤ x ≤ 4
Set up the integral for, but do not calculate, the average value of the given function on the given interval.


Write down the sum that will estimate the given integral using the method of rectangles with mesh of size k . You need not actually evaluate the sum.

Do each of the problems in Exercise 11 with “method of rectangles” replaced by “trapezoid rule.”

Do each of the problems in Exercise 11 with “method of rectangles” replaced by “Simpson’s Rule.”
Answers

At position x in the base circle, the y coordinate is . Therefore the halfdisk slice has radius and area π (1 − x ^{2} )/2. The volume of the solid is then

We calculate the volume of half the solid, and then double the answer. For , at position x in the base square, the y coordinate is . Thus the disk slice has radius and area . Thus the volume of the solid is



The depth of points in the window ranges from 6 to 10 feet. At depth x in this range, the window has chord of length . Thus the total pressure on the lower half of the window is

At depth x , the corresponding subtriangle has sidelength . Therefore the total pressure on one end of the pool is

Let t = 0 be the moment when the climb begins. The weight of the sack at time t is then 100 − t pounds. Then the work performed during the climb is
Thus

The work performed is




We do (a) and (b).

We do (a) and (b)
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From Calculus Demystified: A SelfTeaching Guide. Copyright © 2003 by The McGrawHill Companies, Inc. All Rights Reserved.