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

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?