# Convert the triple integral from rectangular coordinates to both cylindrical and spherical coordinates

The given integral is: $$\int_{-7}^7\int_{-\sqrt{49-x^2}}^\sqrt{49-x^2}\int_{x^2+y^2}^{49}xdxdydx$$ Converting to cylindrical coordinates: $$\int_{0}^{2\pi}\int_{0}^7\int_{r^2}^{49}r^2\cos(\theta)dxdrd\theta$$ It's the spherical coordinates I'm having trouble converting too because I'm don't know how to sketch the given limits. I know that one of the equations is $$x^2+y^2+z^2 = 49$$ but because the limit of $$\theta$$ is $$2\pi$$ I am lost and the example my professor did only went to $$\frac{\pi}{2}$$. I can't find $$\rho$$ because I don't know where the two shapes intersect. I calculated the triple integral for cylindrical coordinates and got zero so I'm very confused.

If the integral is

$$\displaystyle \int_{-7}^7\int_{-\sqrt{49-x^2}}^\sqrt{49-x^2}\int_{x^2+y^2}^{49} x \, dz \, dy \, dx$$

We are basically integrating over the region bound between paraboloid $$z = x^2 + y^2$$ and plane $$z = 49$$.

It is a bit complicated in spherical coordinates and you have to split it into two regions.

$$x = \rho \cos \theta \sin \phi$$
$$y = \rho \sin \theta \sin \phi$$
$$z = \rho \cos \phi$$

In spherical coordinates, $$\phi$$ is the angle made with $$z$$ axis so at $$z = 49$$, $$\phi$$ varies from $$0$$ to $$\cot \phi = \frac{49}{7} = 7 \implies \cot^{-1}(7)$$ with no change in value of $$z$$.

Also as $$z$$ is constant, $$\rho$$ as a function of $$\phi$$ will be $$0 \leq \rho \leq 49 \sec \phi$$ (from equation of $$z$$).

Now for $$\cot^{-1}(7) \leq \phi \leq \frac{\pi}{2}$$, we are traversing till the paraboloid boundary

So, $$z = x^2 + y^2 \implies \rho \cos \phi = \rho^2 \sin^2 \phi \implies \rho = \cot \phi \csc \phi$$

So, $$I = \iiint \rho \cos \theta \sin \phi.\rho^2 \sin \phi \, dr d\phi d \theta$$

with two integrations in two parts with limits of integration -

i) $$0 \leq \rho \leq 49 \sec \phi, 0 \leq \phi \leq \cot^{-1}(7), 0 \leq \theta \leq 2\pi$$.

ii) $$0 \leq \rho \leq \cot \phi \csc \phi, \cot^{-1}(7) \leq \phi \leq \frac{\pi}{2}, 0 \leq \theta \leq 2\pi$$.