15.6 Videos Guide

15.6a

• Notation for triple integrals

  • $\iiint^{\ }_{B}{f(x,\ y,\ z)\ dV}=\int^{f}_{e}{\int^{d}_{c}{\int^{b}_{a}{f(x,\ y,\ z)}\ dxdydz}}$ for
    $B=\left\{ (x,\ y,\ z)\middle|a\leq x\leq b,\ c\leq y\leq d,\ e\leq z\leq f\right\}$
    (a rectangular box—the 2-D trace is $R$)

  • $\iiint^{\ }_{E}{f(x,\ y,\ z)\ dV=\int^{b}_{a}{\int^{h_{2}(x)}_{h_{1}(x)}{\int^{u_{2}(x,\ y)}_{u_{1}(x,\ y)}{f(x,\ y,\ z)\ dzdydx}}}}$ for

$E=\{(x.\ y,\ z)|a\leq x\leq b,\ h_{1}(x)\leq y\leq h_{2}(x),\ u_{1}(x,\ y)\leq z\leq u_{2}(x,\ y)\}$
(a bounded region in $\mathbb{R}^{3}$—the 2-D trace is $D$)

Exercises:

• Evaluate the iterated integral.
  $$\int^{1}_{0}{\int^{1}_{0}{\int^{\sqrt{1-z^{2}}}_{0}{\frac{z}{y+1}\ dx\ dy\ dz}}}$$

15.6b

• Evaluate the triple integral.

  • $\iiint^{\ }_{E}{(x-y)\ dV}$, where $E$ is enclosed by the surfaces $z=x^{2}-1$, $z=1-x^{2}$, $y=0$, and $y=2$

  • $\iiint^{\ }_{E}{z\ dV}$, where $E$ is bounded by the cylinder $y^{2}+z^{2}=9$ and the planes $x=0$,
    $y=3x$, and $z=0$ in the first octant

15.6c

• Use a triple integral to find the volume of the solid enclosed by the paraboloids
  $y=x^{2}+z^{2}$ and $y=8-x^{2}-y^{2}$

15.6d

• Sketch the solid whose volume is given by the iterated integral.
  $$\int^{2}_{0}{\int^{2-y}_{0}{\int^{4-y^{2}}_{0}{dx\ dz\ dy}}}$$

15.6e

• The figure shows the region of integration for the integral $\int^{1}_{0}{\int^{1-x^{2}}_{0}{\int^{1-x}_{0}{f(x,\ y,\ z)\ dy\ dz\ dx}}}$
  Rewrite this integral as an equivalent iterated integral in the five other orders.