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4.5. Homework Problems

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

\[ \int_0^1 e^{x^2} . \]

2. Evaluate the integral

\[ \int_0^{2\pi} \sin x dx , \]

and draw a sketch to explain why your result comes out the way it does.

3. Sketch the graph that represents the definite integral

\[ \int_0^2 (-x^2+2x)dx , \]

and estimate the result roughly from the graph. Then evaluate the integral exactly, and check against your estimate.

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.

5. Show that the mean value theorem's assumption of continuity is necessary, by exhibiting a discontinuous function for which the theorem fails.

6. Show that the fundamental theorem of calculus's assumption of continuity for \(\dot{x}\) is necessary, by exhibiting a discontinuous function for which the theorem fails.

7. Sketch the graphs of y=x2 and \(y=\sqrt{x}\) for 0≤ x≤ 1. Graphically, what relationship should exist between the integrals \(\int_0^1 x^2dx\) and \(\int_0^1 \sqrt{x}dx\) ? Compute both integrals, and verify that the results are related in the expected way.

8. Evaluate \(\int\sqrt{bx\sqrt{x}}dx\), where b is a constant.

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.

hw-piston
Figure K. Problem 9.

10. A certain variable x varies randomly from -1 to 1, with probability distribution dP/dx = k(1-x2).
(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.

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

hw-holditch
Figure I. Problem 13.

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.)