# Untitled Page 23

- Page ID
- 148363

## Homework Problems

**1**.
Write a computer program similar to the one in example 53 on page 74
to evaluate the definite integral

(solution in the pdf version of the book)

**2**.
Evaluate the integral

and draw a sketch to explain why your result comes out the way it does. (solution in the pdf version of the book)

**3**.
Sketch the graph that represents the definite integral

and estimate the result roughly from the graph. Then evaluate the integral exactly, and check against your estimate. (solution in the pdf version of the book)

**4**.
Make a rough guess as to the average value of sin *x* for 0<x<π, and then
find the exact result and check it against your guess.
(solution in the pdf version of the book)

**5**.
Show that the mean value theorem's assumption of continuity is necessary, by exhibiting
a discontinuous function for which the theorem fails.
(solution in the pdf version of the book)

**6**.
Show that the fundamental theorem of calculus's assumption of continuity for x^{·} is necessary, by exhibiting
a discontinuous function for which the theorem fails.
(solution in the pdf version of the book)

**7**.
Sketch the graphs of *y*=*x*^{2} and y=√x for 0≤ *x*≤ 1. Graphically, what relationship
should exist between the integrals \int_{0}^{1} x^{2}dx and \int_{0}^{1} √xdx? Compute both
integrals, and verify that the results are related in the expected way.

**8**.
Evaluate \int√bx√{x}dx, where *b* is a constant.
(solution in the pdf version of the book)

**9**.
In a gasoline-burning car engine, the exploding air-gas
mixture makes a force on the piston, and the force tapers off as the piston expands, allowing the
gas to expand. (a) In the approximation *F*=*k*/*x*, where *x* is the position of the piston, find the
work done on the piston as it travels from *x*=*a* to *x*=*b*, and show that the result only depends
on the ratio *b*/*a*. This ratio is known as the compression ratio of the engine. (b) A better
approximation, which takes into account the cooling of the air-gas mixture as it expands, is
*F*=*kx*^{-1.4}. Compute the work done in this case.

**10**.
A certain variable *x* varies randomly from -1 to 1, with probability distribution
*dP*/*dx*=*k*(1-*x*^{2}).

(a) Determine *k* from the requirement of normalization.

(b) Find the average value of *x*.

(c) Find its standard deviation.

**11**.
Suppose that we've already established that the derivative of an odd function is even, and vice versa.
(See problem 30, p. 50.) Something similar can be proved for integration.
However, the following is not quite right.

*Let f be even, and let g=\int f(x)dx be its indefinite integral. Then by the fundamental theorem
of calculus, f is the derivative of g. Since we've already established that the derivative of an odd
function is even, we conclude that g is odd.*

Find all errors in the proof. (solution in the pdf version of the book)

**12**.
A perfectly elastic ball bounces up and down forever, always coming back up to the same height *h*. Find
its average height.

**13**.
The figure shows a curve with a tangent line segment of length 1 that sweeps around it, forming a new curve
that is usually outside the old one.
Prove Holditch's theorem, which states that the new curve's area differs from the old one's
by π.
(This is an example of a result that is much more
difficult to prove without making use of infinitesimals.)

(c) 1998-2013 Benjamin Crowell, licensed under the Creative Commons Attribution-ShareAlike license. Photo credits are given at the end of the Adobe Acrobat version.