$next$ $up$ $previous$
Next: About this document ... Up: Labs and Projects for Previous: Definite Integrals

Exercises

1.

This exercise emphasizes that the value of a definite integral is found as the limit of Riemann sums. Use the function g defined in the background section.

(a): Run the commands given in the background section to obtain a right sum and find its limit.
(b): Follow the pattern of the sample commands to obtain a left sum and find its limit.
(c): Follow the pattern of the sample commands to obtain a middle point sum and find its limit.
(d): What theorem in the text guarantees that the limits in (a), (b) and (c) are the same? What property of g makes it possible to apply that theorem here? Comment on your answer.

2.

Use the int command to try to find antiderivatives for the following functions. What important part is missing from the answers that Maple provided?

(a): $\begin{maplelatex} \begin{displaymath} \cos^4(x)\end{displaymath}\end{maplelatex}$
(b): $\begin{maplelatex} \begin{displaymath} \displaystyle\frac{x}{x^6 + 1}\end{displaymath}\end{maplelatex}$
(c): $\begin{maplelatex} \begin{displaymath} \sin(x^2)\end{displaymath}\end{maplelatex}$
This problem shows that Maple cannot always find a ``nice'' antiderivative. The problem may not be with Maple; some functions do not have usable antiderivatives.

3.

Use the int command to try to evaluate the following definite integrals.

(a): $\begin{maplelatex} \begin{displaymath} \displaystyle\int^\pi_0 \cos^4(x)dx\end{displaymath}\end{maplelatex}$
(b): $\begin{maplelatex} \begin{displaymath} \displaystyle\int^4_1 \displaystyle\frac{x}{x^6 + 1}\;dx\end{displaymath}\end{maplelatex}$
The answer is complicated looking. Use evalf to approximate it.
(c): $\begin{maplelatex} \begin{displaymath} \displaystyle\int^\pi_0\sin(x^2)dx\end{displaymath}\end{maplelatex}$
Use evalf to approximate the odd looking answer.

4.

Recall that $\displaystyle\frac{d}{dx}(\tan(x)) = \sec^2(x).$

(a): Consider $\int \tan(x)\sec^2(x)dx$ . Note that this has the form $\int u du$ with $u = \tan(x)$ . Find the antiderivative by hand (not by Maple).
(b): Use Maple to evaluate $\int \tan(x)\sec^2(x)dx$ .
(c): Since the answers in (a) and (b) are not the same, who is wrong? Can it be that both you and Maple are correct? Explain what is happening.

5.

Consider $g(x) = -\displaystyle\frac{1}{12}x^4+2x^2-3x+5$ on the interval [-1,2].

(a): Use Maple to set up M₆. Use evalf to evaluate the expression Maple gives.
(b): If M₆ is used to approximate $\int^2_{-1} g(x)d(x)$ , what accuracy is guaranteed by the error bound formula? That is, what is the greatest amount by which M₆ might differ from $\int^2_{-1}g(x)dx$ ? (This part of the exercise can be done by hand on a sheet attached at the back of the Maple pages.)
(c): Use (a) and (b) to give an interval in which the number $\int^2_{-1}g(x)dx$ must be contained.
(d): In fact, how good an approximation is M₆? Use int to get the exact value of $\int^2_{-1}g(x)dx$ . (You should not have done this prior to now.) Compute
$\mid \int^2_{-1}g(x)dx - M_6\mid$ . How does this number compare with your answer in (b)? Explain why the two numbers differ.

$next$ $up$ $previous$
Next: About this document ... Up: Labs and Projects for Previous: Definite Integrals

Christine Marie Bonini
4/5/1999