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Definite Integrals

Definite Integrals

At a Glance - How to Solve a Math Problem

  • Figure out what the problem is asking.
  • Solve the problem.
  • Check the answer.

Sample Problem

Find  given that f is odd and

Answer:

  • Figure out what the problem is asking.

We need to use various properties of integrals to rearrange this equation until there's a  in it somewhere. Then we solve for 

  • Solve the problem.

There are so many ways to start this, deciding which to do is a project in itself. We'll do one way to get the answer here, and we'll do it a different way to check our work.

First, split up the integral to get

Work with the first term first. Since the first term is the integral of the constant 4 on an interval of length 12, that term is equal to 48.

Now we have

Switch the limits and the sign:

Pull out the constant:

Since f is odd, , so

Now we have

We know this is supposed to be equal to 3:

Solving,

.

  • Check the answer.

If we do the problem again, but a different way, we should get the same answer.

Starting with

switch the limits and the sign first this time:

Now we'll split up the integral, but we need to be careful that the negative sign still affects everything:

Since f is odd, 3f(x) is also odd. This means , since

 is zero. Put this back into our expression:

.

Now we pull out the constant:

We can substitute 48 for  like we did before:

It's time to start solving. We know this is all supposed to equal 3.

Adding 3 and  to both sides,

and again we get

which is reassuring.

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