5.4 Triple integrals  (Page 7/8)

 Page 7 / 8

Set up the integral that gives the volume of the solid $E$ bounded by $x={y}^{2}+{z}^{2}$ and $x={a}^{2},$ where $a>0.$

Find the average value of the function $f\left(x,y,z\right)=x+y+z$ over the parallelepiped determined by $x=0,x=1,y=0,y=3,z=0,$ and $z=5.$

$\frac{9}{2}$

Find the average value of the function $f\left(x,y,z\right)=xyz$ over the solid $E=\left[0,1\right]\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\left[0,1\right]\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\left[0,1\right]$ situated in the first octant.

Find the volume of the solid $E$ that lies under the plane $x+y+z=9$ and whose projection onto the $xy$ -plane is bounded by $x=\sqrt{y-1},x=0,$ and $x+y=7.$

$\frac{156}{5}$

Find the volume of the solid E that lies under the plane $2x+y+z=8$ and whose projection onto the $xy$ -plane is bounded by $x={\text{sin}}^{-1}y,y=0,$ and $x=\frac{\pi }{2}.$

Consider the pyramid with the base in the $xy$ -plane of $\left[-2,2\right]\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\left[-2,2\right]$ and the vertex at the point $\left(0,0,8\right).$

1. Show that the equations of the planes of the lateral faces of the pyramid are $4y+z=8,$ $4y-z=-8,$ $4x+z=8,$ and $-4x+z=8.$
2. Find the volume of the pyramid.

a. Answers may vary; b. $\frac{128}{3}$

Consider the pyramid with the base in the $xy$ -plane of $\left[-3,3\right]\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\left[-3,3\right]$ and the vertex at the point $\left(0,0,9\right).$

1. Show that the equations of the planes of the side faces of the pyramid are $3y+z=9,$ $3y+z=9,$ $y=0$ and $x=0.$
2. Find the volume of the pyramid.

The solid $E$ bounded by the sphere of equation ${x}^{2}+{y}^{2}+{z}^{2}={r}^{2}$ with $r>0$ and located in the first octant is represented in the following figure.

1. Write the triple integral that gives the volume of $E$ by integrating first with respect to $z,$ then with $y,$ and then with $x.$
2. Rewrite the integral in part a. as an equivalent integral in five other orders.

a. $\underset{0}{\overset{4}{\int }}\phantom{\rule{0.2em}{0ex}}\underset{0}{\overset{\sqrt{{r}^{2}-{x}^{2}}}{\int }}\phantom{\rule{0.2em}{0ex}}\underset{0}{\overset{\sqrt{{r}^{2}-{x}^{2}-{y}^{2}}}{\int }}dz\phantom{\rule{0.2em}{0ex}}dy\phantom{\rule{0.2em}{0ex}}dx;$ b. $\underset{0}{\overset{2}{\int }}\phantom{\rule{0.2em}{0ex}}\underset{0}{\overset{\sqrt{{r}^{2}-{y}^{2}}}{\int }}\phantom{\rule{0.2em}{0ex}}\underset{0}{\overset{\sqrt{{r}^{2}-{x}^{2}-{y}^{2}}}{\int }}dz\phantom{\rule{0.2em}{0ex}}dx\phantom{\rule{0.2em}{0ex}}dy,$ $\underset{0}{\overset{r}{\int }}\phantom{\rule{0.2em}{0ex}}\underset{0}{\overset{\sqrt{{r}^{2}-{z}^{2}}}{\int }}\phantom{\rule{0.2em}{0ex}}\underset{0}{\overset{\sqrt{{r}^{2}-{x}^{2}-{z}^{2}}}{\int }}dy\phantom{\rule{0.2em}{0ex}}dx\phantom{\rule{0.2em}{0ex}}dz,$ $\underset{0}{\overset{r}{\int }}\phantom{\rule{0.2em}{0ex}}\underset{0}{\overset{\sqrt{{r}^{2}-{x}^{2}}}{\int }}\phantom{\rule{0.2em}{0ex}}\underset{0}{\overset{\sqrt{{r}^{2}-{x}^{2}-{z}^{2}}}{\int }}dy\phantom{\rule{0.2em}{0ex}}dz\phantom{\rule{0.2em}{0ex}}dx,$ $\underset{0}{\overset{r}{\int }}\phantom{\rule{0.2em}{0ex}}\underset{0}{\overset{\sqrt{{r}^{2}-{z}^{2}}}{\int }}\phantom{\rule{0.2em}{0ex}}\underset{0}{\overset{\sqrt{{r}^{2}-{y}^{2}-{z}^{2}}}{\int }}dx\phantom{\rule{0.2em}{0ex}}dy\phantom{\rule{0.2em}{0ex}}dz,$ $\underset{0}{\overset{r}{\int }}\phantom{\rule{0.2em}{0ex}}\underset{0}{\overset{\sqrt{{r}^{2}-{y}^{2}}}{\int }}\phantom{\rule{0.2em}{0ex}}\underset{0}{\overset{\sqrt{{r}^{2}-{y}^{2}-{z}^{2}}}{\int }}dx\phantom{\rule{0.2em}{0ex}}dz\phantom{\rule{0.2em}{0ex}}dy$

The solid $E$ bounded by the sphere of equation $9{x}^{2}+4{y}^{2}+{z}^{2}=1$ and located in the first octant is represented in the following figure.

1. Write the triple integral that gives the volume of $E$ by integrating first with respect to $z,$ then with $y,$ and then with $x.$
2. Rewrite the integral in part a. as an equivalent integral in five other orders.

Find the volume of the prism with vertices $\left(0,0,0\right),\left(2,0,0\right),\left(2,3,0\right),$ $\left(0,3,0\right),\left(0,0,1\right),\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\left(2,0,1\right).$

$3$

Find the volume of the prism with vertices $\left(0,0,0\right),\left(4,0,0\right),\left(4,6,0\right),$ $\left(0,6,0\right),\left(0,0,1\right),\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\left(4,0,1\right).$

The solid $E$ bounded by $z=10-2x-y$ and situated in the first octant is given in the following figure. Find the volume of the solid.

$\frac{250}{3}$

The solid $E$ bounded by $z=1-{x}^{2}$ and situated in the first octant is given in the following figure. Find the volume of the solid.

The midpoint rule for the triple integral $\underset{B}{\iiint }f\left(x,y,z\right)dV$ over the rectangular solid box $B$ is a generalization of the midpoint rule for double integrals. The region $B$ is divided into subboxes of equal sizes and the integral is approximated by the triple Riemann sum $\sum _{i=1}^{l}\sum _{j=1}^{m}\sum _{k=1}^{n}f\left(\stackrel{–}{{x}_{i}},\stackrel{–}{{y}_{j}},\stackrel{–}{{z}_{k}}\right)\text{Δ}V,$ where $\left(\stackrel{–}{{x}_{i}},\stackrel{–}{{y}_{j}},\stackrel{–}{{z}_{k}}\right)$ is the center of the box ${B}_{ijk}$ and $\text{Δ}V$ is the volume of each subbox. Apply the midpoint rule to approximate $\underset{B}{\iiint }{x}^{2}dV$ over the solid $B=\left\{\left(x,y,z\right)|0\le x\le 1,0\le y\le 1,0\le z\le 1\right\}$ by using a partition of eight cubes of equal size. Round your answer to three decimal places.

$\frac{5}{16}\approx 0.313$

[T]

1. Apply the midpoint rule to approximate $\underset{B}{\iiint }{e}^{\text{−}{x}^{2}}dV$ over the solid $B=\left\{\left(x,y,z\right)|0\le x\le 1,0\le y\le 1,0\le z\le 1\right\}$ by using a partition of eight cubes of equal size. Round your answer to three decimal places.
2. Use a CAS to improve the above integral approximation in the case of a partition of ${n}^{3}$ cubes of equal size, where $n=3,4\text{,…,}\phantom{\rule{0.2em}{0ex}}10.$

Suppose that the temperature in degrees Celsius at a point $\left(x,y,z\right)$ of a solid $E$ bounded by the coordinate planes and $x+y+z=5$ is $T\left(x,y,z\right)=xz+5z+10.$ Find the average temperature over the solid.

$\frac{35}{2}$

Suppose that the temperature in degrees Fahrenheit at a point $\left(x,y,z\right)$ of a solid $E$ bounded by the coordinate planes and $x+y+z=5$ is $T\left(x,y,z\right)=x+y+xy.$ Find the average temperature over the solid.

Show that the volume of a right square pyramid of height $h$ and side length $a$ is $v=\frac{h{a}^{2}}{3}$ by using triple integrals.

Show that the volume of a regular right hexagonal prism of edge length $a$ is $\frac{3{a}^{3}\sqrt{3}}{2}$ by using triple integrals.

Show that the volume of a regular right hexagonal pyramid of edge length $a$ is $\frac{{a}^{3}\sqrt{3}}{2}$ by using triple integrals.

If the charge density at an arbitrary point $\left(x,y,z\right)$ of a solid $E$ is given by the function $\rho \left(x,y,z\right),$ then the total charge inside the solid is defined as the triple integral $\underset{E}{\iiint }\rho \left(x,y,z\right)dV.$ Assume that the charge density of the solid $E$ enclosed by the paraboloids $x=5-{y}^{2}-{z}^{2}$ and $x={y}^{2}+{z}^{2}-5$ is equal to the distance from an arbitrary point of $E$ to the origin. Set up the integral that gives the total charge inside the solid $E.$

Questions & Answers

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
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Abhi
how do they get the third part x = (32)5/4
kinnecy Reply
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Jeffrey Reply
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ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
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Commplementary angles
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The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
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At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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