Exercises

1.

Use Maple to evaluate the following integrals.

a.

$${\int x^2 dx}$$

b.

$${\int x^2\cos 2x\,dx}$$

c.

$${\int_0^\pi x^2\cos 2x\,dx}$$

d.

$${\int_{-1}^c \frac{x}{\sqrt{x^2+1}}dx}$$

2.

Recall that $\sin 2x = 2\cos x\,\sin x$ is an identity. That is, the two expressions are mathematically identical. Have Maple find the integral of $\sin 2x$ and the integral of $2\cos x\,\sin x$.Are the two answers mathematically the same? What's going on?

3.

For each of the following plane regions, plot the region involved, and write a Riemann sum that approximates the area. Then determine the definite integral that gives the answer. Write the function to be integrated and the limits of integration. Explain the procedure you used to set up the integral. Finally, use Maple to find a numerical value for the area. Is your answer reasonable?

a.

The region between the positive x-axis, the lines x=0 and x=3, and the curve y=(x+1)^-2

b.

The region above the line y=1 and below the curve $y=\frac{1}{2}x^2+2x+3$, and between x=-2 and x=1

c.

The region bounded by x=0, y=x²-x, and $y=\sqrt{1-x}$.

d.

The region below $y=\cos x^2$ and above the interval [-1.0,1.0]. (Note: the integral of $\cos x^2$ is one of the Fresnel integrals, which are important in optics. They cannot be written in terms of elementary functions. Using decimal numbers instead of integers in the limits of integration forces Maple to return a numerical answer.)

e.

The bounded region between the curves y=4x-x² and $y=2\cos x$

4.

If electric power is used at a constant rate R over a time interval of length T, the total energy consumed is RT. Suppose that a certain business's electricity use varies during the day, from a maximum of 1.5 megawatts at noon to a minimum of 0.3 megawatts at midnight. Assume that the dependence is sinusoidal; i.e., the graph is shaped like a sine or cosine curve.

a.

Find a formula for R as a function of time. Plot this function and verify that it has the desired properties. (Hint: you can adjust the period of a cosine curve by writing $\cos \omega t$, where $\omega$ is a constant. How would you adjust the maximum and minimum values of the curve?)

b.

How much electricity is consumed during a 24-hour period? Describe how you obtained your answer. Is your answer reasonable?