Lower Level integration problem (Find the centroid)

[J-villa]

EDIT - Solved. Thank you.

Homework Statement

Find the Centroid of the region bounded by the x-axis and $$\sqrt{9-x^2}$$

Homework Equations

So far, I found my Mx which ended up being 27p after applying this forumula:

p$$\int$$($$\frac{\sqrt{9-x^2}+0}{2}$$) ($$\sqrt{9-x^2}$$-0)dx​

*My interval was [-3,3]

The Attempt at a Solution

My problem is when I go to find the mass so I can find the centroid and get my final answer. The equation to find the mass is:

m=p$$\int$$[f(x)-g(x)]dx

so when applying the formula to my problem, I get this:

m=p$$\int$$$$\sqrt{9-x^2}$$

So really, it ends up being a simple integration stumble. I just can't figure out how to integrate this. I tried the substitution rule, but it didn't work out.

Can someone point me in the right direction please?

 

[elliotician]

I'm guessing you have probably gone wrong somewhere unfortunately i don't know where. But the solution to your integral:

http://www5a.wolframalpha.com/Calculate/MSP/MSP8641961aa713bih43db00000ebbd4gc0a51fd4i?MSPStoreType=image/gif&s=15

 

[J-villa]

Thank you greatly for the reply. I'm in Calc. II, learning all this integration as fast as I can and I stumble here and there in my theorems.

I was looking at my inverse and hyperbolic trig. functions to try and make a connection, but never thought of double substituting regular sine.

However, I'm not the kinda person to write an answer/work down and claim it as my own, especially since I won't be able to do it when the test comes, so using your reply I have a few questions.

• #1 - I notice you didn't see the constant (p) I have to have prior to the integral. I'm guessing that the p would just go along for the ride with the all the other constants.
  
• #2 - speaking of constants, how did you get that 9 outside the integral (Step:1/2)? I was thinking you just pulled the 9 out from the integrand; but wouldn't that leave a 1-sin^2?

Once again, thanks for the responses. I'm going to be lurking on these forums a lot more often now.

 

[elliotician]

We are using the constant multiple rule for integration and 1-sin^2=cos^2.

 

[J-villa]

We are using the constant multiple rule for integration and 1-sin^2=cos^2.

oh, duh'. I appreciate the help. I'll respond soon w/ my answer.

 

[vaibhav1803]

substitute x=3sin(t)..and the answer comes by itself...