Evaluate the definite integral:

[1question]

Homework Statement

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

Homework Equations

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?

 

[Dick]

Homework Statement

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

Homework Equations

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?

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

 

[Sunil Simha]

I am stuck. I tried to do it with with anti-differentiation and it didn't work/very computation-intensive.

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.

 

[1question]

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

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?)

 

[Dick]

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.

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

 

[Dick]

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?)

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.

 

[1question]

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.

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.

 

[Dick]

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.

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.

 

[1question]

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.

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!