How can I find the volume of this strange body?

Question

I need to calculate the volume of a body using multiple integrals. The body's boundary is defined by a surface whose formula is

$$0 \leq z \leq c \cdot \sin\left(\pi \sqrt{{{x^2} \over{a^2}} + {{y^2} \over{b^2}}}\right)$$

where $a$, $b$ and $c$ are some positive parameters.

The problem is that I have no idea what this body looks like. It's the first time I'm meeting sinus function in multitple integrals and it totally confuses me. I've tried to use the most popular substitutes, such as cylindrical and spherical coordinates, but it didn't help at all.

Thanks for any help!

Answer 1

If I understand this problem correctly, it is a matter of recognizing the elliptical coordinates inside the sine. Thus, if we let

$$x=a k \cos{t}$$ $$y=b k \sin{t}$$

where $k \in [0,1]$ and $t \in [0,2 \pi]$. In this way, $z \in [0,c \sin{(\pi k)}]$. Then the integral may be written as

$$a b \int_0^{2 \pi} dt \, \int_0^1 dk \, k \, \int_0^{c \sin{(\pi k)}} dz $$

Note that I made use of the Jacobian to effect the coordinate change.

Answer 2

For $a = b = c = 1$, the surface looks like this

However, the surface intersects the $z = 0$ plane

Hence, the body is not connected. The volume of each component can be computed using polar coordinates. This volume is $\infty$.