What are some good moderately-difficult to difficult multivariable integral problems?

I'm looking for interesting, difficult, or otherwise clever multivariable integral problems that are more difficult than usual textbook problems (which, in the textbook I'm reading at least, usually involve either reordering an iterated integral, or making a fairly usual substitution like polar or spherical coordinates).

Namely, I'm interested in problems that involve either tricky uses of multivariable substitution, interesting interpretations of the problem (i.e. to solve the problem you need to use a multivariable integral, but how?), clever partitioning of the domain of integration, or other interesting maneuvers.

To give some positioning, a problem that would be too easy is to solve the following:

$$\int_0^3\int_{x^2}^9 x^3e^{y^3}\text{d}y\text{d}x$$

And for completeness, here's a problem that would probably be too hard.

Edit: I'm also interested in more obscure and unusual problems.

Edit 2: Here are some more problems I'd consider "too easy":

$$\iiint_S\sin\sqrt{x^2+y^2+z^2}\text{d}V$$ where $S$ is the region bounded by $x^2+y^2+z^2 = 49$ and $z^2=x^2+y^2$.

$$\int_0^2\int_0^1\int_y^1\sinh(z^2)\text{d}z\text{d}y\text{d}x$$

One thing that would be nice are questions that require nonstandard substitutions. Everywhere I look only exercises the cylindrical and spherical coordinate transformations, but Wikipedia has an expansive list of other interesting coordinate systems. What about integration over a torus? Or the intersection of a torus and a hyperbolic paraboloid? What about integrals that require bizarre transformations to complete, ones that don't even have names?

I want problems that really exercise one's ability to decipher the best solution to the integral, to understand a difficult region of integration geometrically, and/or to call from different areas of mathematics to solve the integral in unique ways.

By "decipher", I mean "see the trick to dig into a problem and make it easier". For example, the following integral looks ridiculous: $$\int_0^1\int_{2\sqrt x}^{1+x}\frac{x}{y+1}\frac{\text{d}y\text{d}x}{\sqrt{y^2-4x}}$$ If you take the time to experiment, you might find that the substitution $x = uv$, $y = u + v$ turns the integral into $$\int_0^1\int_u^1\frac{uv}{u+v+1}\text{d}v\text{d}u$$ which can be solved more easily.

Disclaimer: this is a very contrived problem, I made it up by starting with the end and working in reverse, but it gives you some idea as to the standard of non-triviality I'm hoping for. In actuality, I would deem this integral uninteresting because there's no "simple but clever/difficult to find" trick.

• This isn't an integral problem, but it's still an interesting problem for students in a multivariable calculus class. Prove AM-GM by maximizing $F(x_1,\dots, x_n) = \prod_{i = 1}^n x_i^2$ subject to $\sum_{i = 1}^n x_i^2 = 1$. – Stahl Aug 9 '17 at 22:55
• Long time ago, when I was a student, we used “Problems in mathematical analysis” by Demidovich (English translation is available at LibGen). It contains a huge number (about four and half thousands) of problems and was so popular that there was written a book of its solutions, called Antidemidovich. The problems are of different hardness. On the one hand, I guess it was a common student problem book, so it should contain a lot of easy problems. – Alex Ravsky Aug 19 '17 at 19:36
• But, on the other hand, I was one of the strongest students at our flow and I wanted to pass the exams without attending classes, so the dean of our faculty who taught us analysis, created for me lists of problems from this book to look. The last chapter of the book is devoted to multivariable and curvilinear integrals and contains more than a half of a thousand problems, so you may search it for problems suitable for you. – Alex Ravsky Aug 19 '17 at 19:37
• @AlexRavsky This is an amazing resource, thank you! If you have any personal favourites you'd like to share you could post them as an answer. – Michael Ala Aug 19 '17 at 20:02
• Unfortunately, I remember no favorites, because I studied this book twenty years ago. – Alex Ravsky Aug 20 '17 at 5:50

For positive scalars $a_1,\dots,a_n$, define $\Delta^n(a_1,\dots,a_n)$ to be the subset of $\Bbb R^n$ given by $$\Delta^n(a_1,\dots,a_n) = \left\{(x_1,\dots,x_n)\in\Bbb R^n\mid 0\le \frac{x_1}{a_1}+\dotsb+\frac{x_n}{a_n}\le 1\right\}.$$ Show that $$\operatorname{volume}(\Delta^n(a_1,\dots,a_n)) = \frac{a_1\dotsb a_n}{n!}.$$ The region $\Delta^n(a_1,\dots,a_n)$ is known as the $n$-simplex.
Perhaps you've seen this one before; it's a classical example of a tricky integral: $$\int_{-\infty}^\infty e^{-x^2} dx$$ It has no elementary indefinite integral, yet we can evaluate its value on $\mathbb{R}$ through a couple different methods. One method in particular (perhaps more I'm unaware of) makes clever use of multiple integration.