Evaluate the triple integral of sin of (x squared + y squared + z squared) to the power 3/2, over the region between the cone z = root of 3(x squared + y squared) and the sphere z = root of (1 - x squared - y squared).

Step-by-step solution

1. 1

   Identify the solid. The lower surface $z=\sqrt{3}\,r$ (with $r^2=x^2+y^2$) is a cone, and the upper surface $z=\sqrt{1-r^2}$ is the upper unit hemisphere. They meet where $\sqrt3\,r=\sqrt{1-r^2}$, i.e. $3r^2=1-r^2$, so $r=\tfrac12$: the region is the classic 'ice-cream cone' between the cone and the unit sphere, projecting onto the disc $r\le\tfrac12$.

2. 2

   Choose spherical coordinates, not cylindrical. The integrand depends only on $x^2+y^2+z^2 = \rho^2$, so in spherical coordinates it becomes $\sin(\rho^3)$ - a function of one variable. Cylindrical coordinates would leave an inseparable $\sin\!\left((r^2+z^2)^{3/2}\right)$, which is why the conversion has to go all the way to spherical.

3. 3

   Convert the cone into a constant polar angle. On the cone, $\tan\varphi = \dfrac{r}{z} = \dfrac{1}{\sqrt3}$, so $\varphi = \dfrac{\pi}{6}$. The solid is therefore $0\le\rho\le1,\qquad 0\le\varphi\le\frac{\pi}{6},\qquad 0\le\theta\le2\pi,$ with volume element $dV=\rho^2\sin\varphi\,d\rho\,d\varphi\,d\theta$.

4. 4

   Separate the triple integral into three single ones. $I = \int_0^{2\pi}\!d\theta\int_0^{\pi/6}\!\sin\varphi\,d\varphi\int_0^{1}\rho^2\sin(\rho^3)\,d\rho.$ Each factor now involves a single variable.

5. 5

   Evaluate each factor. $\int_0^{2\pi}d\theta = 2\pi$; $\int_0^{\pi/6}\sin\varphi\,d\varphi = 1-\cos\tfrac{\pi}{6} = 1-\tfrac{\sqrt3}{2}$; and with $u=\rho^3$, $du=3\rho^2 d\rho$, $\int_0^1\rho^2\sin(\rho^3)\,d\rho = \frac13\int_0^1\sin u\,du = \frac{1-\cos 1}{3}.$ The $\rho^2$ from the volume element is exactly what makes this substitution work.

6. 6

   Multiply and check. $I = 2\pi\left(1-\frac{\sqrt3}{2}\right)\frac{1-\cos 1}{3} \approx 2\pi(0.133975)(0.153233) \approx 0.128989.$ Nested numerical quadrature of the original cylindrical form gives $0.1289892138$, matching to nine decimals.