1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Evaluate the definite integral:

  1. Apr 22, 2013 #1
    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. jcsd
  3. Apr 22, 2013 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Think about what the graphs of x^101 and sqrt(9-x^2) look like.
     
  4. Apr 22, 2013 #3
    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.
     
  5. Apr 22, 2013 #4
    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?)
     
  6. Apr 22, 2013 #5

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    No, it's not hard to do the integration. But the point here is to find an even easier shortcut.
     
  7. Apr 22, 2013 #6

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    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.
     
  8. Apr 22, 2013 #7
    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.
     
  9. Apr 22, 2013 #8

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    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.
     
  10. Apr 22, 2013 #9
    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!!!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Evaluate the definite integral:
Loading...