3.5 Triple Integrals

Evaluate the integral

without using iteration. Instead, interpret the integral geometrically.

Find the total mass of the rectangular box $[0, 1] \times [0, 2] \times [0, 3]$ (that is, the box defined by the inequalities $0 \le x \le 1$, $0 \le y \le 2$, $0 \le z \le 3$), with density function $h(x, y, z) = x$.

Evaluate $\dst\tripInt_R x\ \dee{V}$ where $R$ is the tetrahedron bounded by the coordinate planes and the plane $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$.

Evaluate $\dst\tripInt_R y\ \dee{V}$ where $R$ is the portion of the cube $0\le x,y,z\le 1$ lying above the plane $y+z=1$ and below the plane $x+y+z=2$.

For each of the following, express the given iterated integral as an iterated integral in which the integrations are performed in the order: first $z$, then $y$, then $x$.

A triple integral $\dst\tripInt_E f\ \dee{V}$ is given in iterated form by

A triple integral $\tripInt_E f(x,y,z)\ \dee{V}$ is given in the iterated form

Write the integral given below $5$ other ways, each with a different order of integration.

Let $\displaystyle I = \tripInt_E f(x,y,z)\ \dee{V}$ where $E$ is the tetrahedron with vertices $(-1, 0, 0)$, $(0, 0, 0)$, $(0, 0, 3)$ and $(0, -2, 0)$.

Let $T$ denote the tetrahedron bounded by the coordinate planes $x = 0$, $y = 0$, $z = 0$ and the plane $x + y + z = 1$. Compute

Let $E$ be the portion of the first octant which is above the plane $z = x + y$ and below the plane $z = 2$. The density in $E$ is $\rho(x, y, z) = z$. Find the mass of $E$.

Evaluate the triple integral $\tripInt_E x\ \dee{V}$, where $E$ is the region in the first octant bounded by the parabolic cylinder $y = x^2$ and the planes $y + z = 1$, $x = 0$, and $z = 0$.

Let $E$ be the region in the first octant bounded by the coordinate planes, the plane $x + y = 1$ and the surface $z = y^2$ . Evaluate $\tripInt_E z\ \dee{V}$ .

Evaluate $\tripInt_R yz^2 e^{-xyz}\ \dee{V}$ over the rectangular box

Let

Express $J$ as an integral where the integrations are to be performed in the order $x$ first, then $y$, then $z$.

Let $E$ be the region bounded by $z = 2x$, $z = y^2$, and $x = 3$. The triple integral $\tripInt f(x,y,z)\,\dee{V}$ can be expressed as an iterated integral in the following three orders of integration. Fill in the limits of integration in each case. No explanation required.

Let E be the region inside the cylinder $x^2 + y^2 = 1$, below the plane $z = y$ and above the plane $z = -1$. Express the integral

as three different iterated integrals corresponding to the orders of integration: (a) $\dee{z}\, \dee{x}\, \dee{y}$, (b) $\dee{x}\, \dee{y}\, \dee{z}$, and (c) $\dee{y}\, \dee{z}\, \dee{x}$.

Let $E$ be the region bounded by the planes $y=0$, $y=2$, $y+z=3$ and the surface $z=x^2$. Consider the intergal

Fill in the blanks below. In each part below, you may need only one integral to express your answer. In that case, leave the other blank.

Evaluate $\tripInt_E z\,\dee{V}$, where $E$ is the region bounded by the planes $y=0$, $z=0$ $x+y=2$ and the cylinder $y^2+z^2=1$ in the first octant.

Find $\dst\tripInt_D x\,\dee{V}$ where $D$ is the tetrahedron bounded by the planes $x=1$, $y=1$, $z=1$, and $x+y+z=2$.

The solid region $T$ is bounded by the planes $x=0$, $y=0$, $z=0$, and $x+y+z=2$ and the surface $x^2+z=1$.