How to evaluate multi-variable integrals such as $\int xy\, dx\,dy$?

Question

How are integrals with two variables calculated?

Like $$\int xy\, dx\,dy$$

I'm new in multivariable calculus so please help.

Answer 1

I am a bit confused about your notation. The normal way to denote what you intended is use the number of integrals corresponding to the number of variables (in your case 2), which in LaTeX can be done with \iint for 2 or \iiint for 3 variables.

The usual idea is to compute double integral as an integral of an integral, e.g. $$ \begin{split} \iint xy\ dx dy &= \int \left[\int xy dy\right] dx \\ &= \int x \left[\int y dy\right] dx \\ &= \int x \left[\frac{y^2}{2}\right] dx \\ &= \frac{y^2}{2} \int x dx \\ &= \frac{x^2 y^2}{4} + C \end{split} $$

Now that you understand that double integral is really an integral of an integral, the $\iint$ notation also makes sense.

Note that this goes $y$-first, but going $x$-first would not change things. Formally, you need Fubini's Theorem to know that though...

UPDATE

As is explicitly pointed out in the comments, $C$ in case of one-variable integration amounts to a constant, here would have the form $C(y) + K(x)$.