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Mathematical Expression Editor

Exercises relating to various topics we have studied.

A key skill to practice is deciding which of the several techniques we have
learned should be applied to a particular problem. In these exercises and
questions, you will be responsible for choosing a method which works for
you.

Compute the indefinite integral. (You should choose an answer which equals at ; if
your answer does not satisfy this, add a constant to make it so.)

Integrate by
parts.

Let and .

Compute the indefinite integral. (You should choose an answer which equals at ; if
your answer does not satisfy this, add a constant to make it so.)

Partial fractions
won’t help in this case because the expression is already simplified as much as
possible.

Try a trigonometric substitution

Make the substitution .

You should arrive at the integral after the substitution; here you’ll need to use a
power-reduction formula to continue.

Sample Quiz Questions

The region in the plane between the -axis and the graph in the range is revolved
around the axis . Compute the volume of the resulting solid.

If the variable is used for slicing, then slices are parallel to the axis of rotation,
which indicates the shell method should be used. The radius of a shell is . The
height of a shell is exactly . The volume of the region is therefore given by
To compute the integral, we can use integration by parts. A reasonable
strategy is to integrate and differentiate . This gives the equality Therefore

Consider the region given by and . Compute the -coordinate of the centroid (i.e.,
assuming constant density).

The mass will be given by the integral One can check that To compute the
-coordinate of the centroid, we also need to compute the integral To compute the
integral, we can use integration by parts. A reasonable strategy is to integrate and
differentiate . This gives the equality Therefore The corret answer is the ratio of the
integrals, i.e.,

Sample Exam Questions

An object moves in such a way that its acceleration at time seconds is meters per
second squared. If the initial velocity of the object is meters per second, what is the
limit of its velocity as ?

meters per second meters per second meters per
second meters per second meters per second meters per second

Find the volume of the solid generated by revolving the region bounded above by
and bounded below by for about the line .

none of these

Evaluate .

none of these

This integral can be computed via integration by parts. If we integrate and
differentiate , we get The latter integral can be simplified using polynomial long
division: . Therefore