# Evaluate the definite integral:

1. Apr 22, 2013

### 1question

1. The problem statement, all variables and given/known data

Evaluate the definite integral ∫(x101-√(9-x^2))dx from -3 to 3.
Hint: This problem can be done without anti-differentiation.

2. Relevant equations

3. The attempt at a solution

I am stuck. I tried to do it with with anti-differentiation and it didn't work/very computation-intensive.
Hints on how to do it without anti-differentiation, please?

Last edited: Apr 22, 2013
2. Apr 22, 2013

### Dick

Think about what the graphs of x^101 and sqrt(9-x^2) look like.

3. Apr 22, 2013

### Sunil Simha

Actually, using indefinite integrals and then applying limits of integration is a pretty good way of solving it. Integral of x101 is easy to calculate and it gets simple to integrate the other term when you substitute x = 3sinθ.

4. Apr 22, 2013

### 1question

Well, I could use the fact that the - and + parts have identical areas (and thus find 1 and multiply by 2 to get the total area?)

5. Apr 22, 2013

### Dick

No, it's not hard to do the integration. But the point here is to find an even easier shortcut.

6. Apr 22, 2013

### Dick

Treat the two function x^101 and sqrt(9-x^2) differently. Start with x^101. What do you say about the relation between the + part and the - part in this case? Are they really the same? For sqrt(9-x^2) sketch a graph. It might be a common geometric shape that you know the area of.

7. Apr 22, 2013

### 1question

They have the same magnitude, but opposite sign.

Well, I know that sqrt(9-x^2) is a semi-circle at x=+/- 3 and height = 3.

8. Apr 22, 2013

### Dick

Good! You've basically got it. If you add "same magnitude, but opposite sign" what do you get? And the integral is the same as the area under the curve, right? So if you know a formula for the area of a semicircle, you know the integral.

9. Apr 22, 2013

### 1question

0?

Oh...

I got it (confirmed with Wolfram) - first term = 0, and the second = πr^2/2 (b/c semi-circle), and so the answer is:

= ...
= (0-πr^2/2)
= 0-π(3)^2/2
= -9π/2

Thank you!!!