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

Here is what I have:

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

[tex]f\left(x, y\right) = x\;y = \left(r\;\cos\theta\right)\left(r\;\sin\theta\right)[/tex]

The area of one quarter of a unti circle is [tex]\frac{\pi}{4}[/tex], right?

[tex]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}[/tex]

Is this correct?