How to separate a double integral function f(x,y)

Question

So, I have integral function $f(x,y)=\int_{-2}^2\int_0^3\sqrt{(4-x^2)(9-y^2)}dydx$ that I want to solve.

Is there a more conventional (or correct, even) way of doing it than separating them as $\int_{-2}^2\sqrt{(4-x^2)}dx\int_{0}^3\sqrt{(9-y^2)}dy$,

because this right here generates a nasty integrals and I think they should generate a more fine mid-function answer.

Answer 1

Is there a more conventional (or correct, even) way of doing it

No.

The way you started is perfectly fine and is the ideal way to do it anyway. Why is it ideal? Recall what a definite integral represents geometrically: area under a curve.

What does the graph of $y = \sqrt{4-x^2}$ look like? It's the upper half of a circle of radius $2$ centered at the origin. So what does $\displaystyle \int_{-2}^2 \sqrt{4-x^2} \, dx$ represent?

Can you take it from here?