Average value of f(x,y) = xy in quarter circle x^2 + y^2 < 1 in Q1.

[VinnyCee]

Here is the problem:

Find the average value of $$f\left(x, y\right) = x\;y$$ for the quarter circle $$x^2 + y^2 \le 1$$ in the first quadrant.

Here is what I have:

Average value equation is $$\frac{1}{Area\;of\;R} \iint_{R} f\left(x, y\right) dA$$

$$f\left(x, y\right) = x\;y = \left(r\;\cos\theta\right)\left(r\;\sin\theta\right)$$

The area of one quarter of a unti circle is $$\frac{\pi}{4}$$, right?

$$Average = \frac{4}{\pi}\;\int_{0}^{\frac{\pi}{2}}\int_{0}^{1}\;\left(r\;\cos\theta\right)\left(r\;\sin\theta\right)\;r\;dr\;d\theta = \frac{1}{2\pi}$$

Is this correct?

 

[Theelectricchild]

Absolutely, and good job remembering the jacobian in polar coordinates.

 

[VinnyCee]

Thank you

Many thanks. I will probably be checking a few others before I am through.

I wonder if I should make a thread for "Check my Calculus Answers Please" and put them all in there or something? On the other hand, I think that individually posting each problem gets quicker results and is easier to search (i guess).

 

[Theelectricchild]

In order to get useful feedback and responses, it's best to title your threads appropriately--- putting something like "check my work..." generally isn't too appealing, so you're probably right--- that is, make threads that are individual for a specific problem, with the statement and reasoning placed in the first post.

You have been doing this but just so others know!