Calculus 3: Triple integrals

Evaluate the triple integral $\int_{-2}^{3} \int_{0}^{2} \int_{0}^{1} x^3 y z^2 \, dz \, dx \, dy$.

Evaluate the integral $\displaystyle \int_{0}^{3} \int_{0}^{x} \int_{0}^{x-y} 4xy \, dz \, dy \, dx$.

Find the bounds for the triple integral in rectangular coordinates using the method of collapsing, for the region bounded by the surfaces: the plane $z = y + 1$, the parabolic cylinder $z = x^2 + 1$, and the plane $y = 1$.

Integrate the function $x + y + z$ from 0 to 1 with respect to $x$, then from 0 to 2 with respect to $z$, and finally from 0 to 3 with respect to $y$.

Calculate the definite integral of the function $x$ from 0 to $\sqrt{xy}$ with respect to $z$, then from 0 to $x$ with respect to $y$, and finally from 0 to 1 with respect to $x$.

Calculate the triple integral of the given region using spherical coordinates, where the region is bounded by a cone and a sphere.

Integrate the region described in spherical coordinates where $\rho = 1$, $0 \leq \phi \leq \frac{\pi}{6}$, and $0 \leq \theta \leq \frac{3\pi}{2}$.

Calculate the volume of a truncated wedge with dimensions: 2 units high, 5 units at the end, 6 units long, and 4 units wide, using a triple integral in rectangular coordinates.

Evaluate the triple integral from 0 to $\pi$ and 0 to 2 and then 0 to $\sqrt{4 - Z^2}$ of $Z \sin Y$ with respect to $dX$ $dZ$ $dy$.

Evaluate the triple integral: $\int_{2}^{4} \int_{-1}^{1} \int_{0}^{3} (y - xz) \, dz \, dy \, dx$.