Calculus 2 : Finding Integrals

Example Questions

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Evaluate:

Explanation:

,

so

Example Question #1 : Finding Integrals

Evaluate:

The integral is undefined.

Explanation:

Rewrite this as follows:

Substitute . Then  and , and the bounds of integration become 2 and 3, making the integral equal to

Example Question #1 : Finding Integrals

Evaluate:

The integral is undefined.

Explanation:

Example Question #1 : Finding Integrals

Evaluate:

Explanation:

Substitute ; so  and , and the bounds of integration become 2 and ; the above becomes

Example Question #1 : Finding Integrals

Which of the following functions makes the statement  true?

Explanation:

Therefore, we are looking for a value of  for which

, or, equivalently,

, or

The only choice that makes  an element of this set is

.

Example Question #1 : Finding Integrals

Evaluate:

Explanation:

An easy way to look at this is to note that on the interval , the integrand

can be rewritten as

Therefore,

The antiderivative of  is . We can evaluate  at each boundary of integration:

Then

The original integral can be evaluated as

Example Question #2 : Finding Integrals

Evaluate:

Explanation:

We evaluate

The original double integral is now

Example Question #2 : Finding Integrals

Evaluate:

Explanation:

We evaluate

The original double integral is now

Example Question #9 : Finding Integrals

Evaluate:

Explanation:

We evaluate

The original double integral is now

Example Question #1 : Finding Integrals

Evaluate:

Explanation:

The problem is easier if it is written as follows:

We evaluate

The original double integral is now

which, similarly to , is equal to 1.

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