How to evaluate $\int_0^3\int_2^0xy\,\mathrm{d}x\,\mathrm{d}y$?

Question

The surface integral is $$\int_0^3\int_2^0xy\,\mathrm{d}x\,\mathrm{d}y$$ where $y=x$. My doubt is, which variable to integrate first $y$ or $x$? When to substitute $y=x$? I used an online integral calculator to evaluate this integral the answer turns out to be $-18$. But If I integrate both variables seperately without substituting $y=x$, I am getting $-9$. I really dont know how to do this integral, please help me by giving a hint to solve this integral.

Answer 1

In general, you must specify which integral corresponds to which variable, but in this case it really doesn't matter. Here is the solution: $$\begin{align}\int_0^3\int_2^0xydxdy&=\int_0^3\left.\left(\frac{x^2}2\right)\right|_2^0ydy\\&=-2\int_0^3ydy\\&=-2\left.\left(\frac{y^2}2\right)\right|_0^3\\&=-2\cdot\frac92\\&=-9\end{align}$$

Edit

I've already explained it in my question. When there are nested integral (like these), you must specify which integral corresponds to which variable. For example $\displaystyle\int_{y=0}^3\int_{x=2}^0xydxdy$. Now there is no confusion.

Notice that the order of integration doesn't matter. It means that

$$\int_{y=0}^3\int_{x=2}^0xydxdy=\int_{x=2}^0\int_{y=0}^3xydxdy$$

Integral variable cannot be a function of something other at the same time. It is either an integral variable, or a function of $x$. If you, for example, know that for every $y$ in that interval, $y=x^2+2$, then you basically have only one integral (replace $y$ with $x^2+2$).