** Next:** About this document ...
**Up:** Labs and Projects for
** Previous:** Labs and Projects for

- Background
- For Exercise 1 .....
- For Exercise 2 .....
- For Exercise 3 .....
- For Exercise 4 .....
- Exercises

In Sections 5.8 and 8.1 you are asked to find antiderivatives by
substitution. This is really a ``change of variables'' technique used
to better see how the integrand can be looked at as the derivative of
some function. Maple has a command called `changevar` that can be
used to practice substitution. This command has three arguments. In
the first argument the quantity to be replaced is followed by an
equals sign which is followed by what will be substituted. The second
argument tells where the substitution will be made and the third
argument gives the variable in terms of which the changed expression
will be given. Note that when an integral in *x* is rewritten with
`changevar`, the *dx* must also be changed, and in a definite
integral, the limits of integration also need to the changed. Since
`changevar` is in the student package, that package must be loaded
before using `changevar`.

Here is an indefinite integral example of the use of `changevar`.
We use `Int` instead of `int` because we do not want the
integral evaluated immediately; we want to be able to observe the
substitution process. After we have done that, we use `value` to
get the answer. Then we use `changevar` one more time in order to
express the answer in terms of the original variable of the problem.
Keep in mind that the aim in doing a substitution is to simplify the
integral to some pattern with which we can readily deal.

> with(student):

> g:=x->x*sqrt(1+x^2);

> I1:=Int(g(x),x);

> changevar(1+x^2=u,I1,u);

> value(``);

> changevar(u=1+x^2,'',x);When you run these commands, notice that the integral is much simplified after the first change of variables. By the way, can you explain why the appears in the output at that time?

Next we look at the commands for a definite integral.

> I2:=Int(g(x),x=2..3);

> changevar(1+x^2=t,I2,t);

> value(``);

In this case we did not have to, in rewriting the integral, switch back
to *x* after integrating because the limits of integration were
switched from limits on *x* to limits on *t*. Make sure you
understand how the limits on *t* were obtained.

You will find the background you need in a section of the text we have covered. Make sure that the answer you give is a reasonable one.

Here is a reminder that should be useful in this exercise. Let us say
we need to find the roots of a polynomial function *h*(*x*). Of
course, this can be done with `fsolve`. However if you name the
output of `fsolve`, then you can have Maple
give you each of the individual roots for further computation. For instance if

> root:=fsolve(h(x)=0,x);then

Volumes of solids of revolution are often found by using the
disk/washer method. (See Section 6.2 of the text.) The Maple `
CalcP` package has three commands that can help you in your study of
this method. The command `revolve` gives a picture of the solid
of revolution being considered. The `LeftDisk` command uses a
regular partition with *x ^{*}*

The third command is `LeftInt`. This evaluates the Riemann sum
associated with the picture given by `LeftDisk`. Note that the
third argument in each of these specifies the numbers of subintervals.

Running the following code will help you see how these commands work

> with(CalcP):

> f:=x->x^2+1;

> plot(f(x),x=-2..2);

> revolve(f(x),x=-2..2);

> revolve(f(x),x=-2..2,y=-2);

> revolve({5,f(x)},x=-2..2);

> LeftDisk(f(x),x=-2..2,5);

> LeftDisk(f(x),x=-2..2,10);

> LeftInt(f(x),x=-2..2,5);

> LeftInt(f(x),x=-2..2,10);

> Pi*int((f(x))^2,x=-2..2);

> evalf(``);The fifth line shows how to revolve the given area about a horizontal line other than the

- 1.
- Use a substitution to help evaluate the following integrals.
Follow the patterns given in the background section. Use a
substitution that results in an integral you could easily evaluate by
hand. When you use your substitution, a constant factor shows up in
the transformed integral. Explain where it comes from.
- (a)
- (b)

- 2.
- Consider on the interval
. Use Maple to find the
*c*whose existence is guaranteed by the Mean Value Theorem for Integrals. Also, give the mean value of the function on the given interval. Make sure your answers are reasonable. - 3.
- Find the total units of area enclosed by the graphs of

and

What special care must be taken in this problem? What is the situation that complicates this problem? Explain.

- 4.
- Solids of revolution.
- (a)
- Run the code given in the background section. Reposition each picture so that it looks 3-dimensional. In particular, display the ``hole'' mentioned in the background.
- (b)
- Given the functions
*f*(*x*) = -*x*+ 5^{2}*x*+ 3 and*g*(*x*) =*x*, consider the volume of the solid of revolution formed when the area enclosed by the two functions is rotated about the^{2}*x*-axis.- i.
- Use
`revolve`to picture the solid and`LeftDisk`with*n*=10 to picture the approximating pieces. Show ``good'' matching views of both pictures. - ii.
- Set up and evaluate the integral that gives the volume.
- iii.
- Use
`LeftInt`with*n*= 100 to estimate the volume. Is the number you get here close enough to the number you got in part ii) for you to feel confident that you handled both parts correctly?

- (c)
- Consider the two commands given below. Run them. They give
approximating pieces for two different solids. For each of them, set
up and evaluate the integral that gives the volume in question. In
your words, describe the area being rotated. It would be a good idea
to use
`LeftInt`to see if your results seem reasonable.> LeftDisk(x^2,x=0..3,6,y=10);

> LeftDisk({x^2,0},x=0..3,6,y=10);

4/8/1999