Question 1

Find the center of mass of the solid with density  and the given shape.  ,

Accepted Answer

The answer of Find the center of mass of the...

Question 2

Find the mass of the solid with density  and the given shape.

Accepted Answer

We need to use the formula:
mass = density x volume
The shape given appears to be a cylinder with radius 2 and height 3. Thus, the volume is:
V = πr^2h = π(2)^2(3) = 12π
Substituting density and volume into the formula:
mass = (11eaa948_cd85_aa2c_84bc_e784fb661ebf_TB2342_11) x (12π)
= 11eaa948_cd85_aa2e_84bc_1123f3ad09a3_TB2342_11
Thus, the answer is A.

Question 3

Evaluate the integral  , where Q is the region with z > 0 bounded by  and  .

Accepted Answer

The answer of Evaluate the integral  , where Q...

Question 4

A function  is a pdf on the three-dimensional region Q if  for all  in Q and  . Find k such that  is a pdf on the region bounded by z = 0, z = 3, x = 0, x = 2, y - z = 0 , and y - z = 3.

Accepted Answer

To find the value of k, we need to use the property that the integral of the pdf over the entire region must equal 1. Mathematically, this is written as:

∫∫∫Q kxyz dV = 1

where Q is the region bounded by z = 0, z = 3, x = 0, x = 2, y - z = 0, and y - z = 3.

The limits of integration for x, y, and z can be obtained from the region boundaries:

0 ≤ x ≤ 2
0 ≤ y - z ≤ 3   (or z ≤ y ≤ z+3)
0 ≤ z ≤ 3

Thus, the integral can be rewritten as:

∫0^3 ∫0^2 ∫z^(z+3) kxyz dy dx dz

Evaluating this integral gives:

∫0^3 ∫0^2 kx/2 [(z+3)^2 - z^2] dx dz
= k/2 ∫0^3 [(z+3)^2 - z^2] ∫0^2 x dx dz
= k/2 ∫0^3 [(z+3)^2 - z^2] dz
= k/2 [∫0^3 (z+3)^2 dz - ∫0^3 z^2 dz]
= k/2 [(1/3)(6^3) - (1/3)(3^3)]
= k/2 (54 - 9)
= 22.5k

Setting this equal to 1, we get:

22.5k = 1
k = 1/22.5
k = 0.0444...

Therefore, the answer is A (0.0444...).

Question 5

Evaluate the triple integral.

Accepted Answer

The answer of Evaluate the triple integral. 
...

Question 6

Evaluate the triple integral  .  ,

Accepted Answer

The answer of Evaluate the triple integral  . ...

Question 7

Evaluate the triple integral  .  ,

Accepted Answer

The answer of Evaluate the triple integral  . ...

Question 8

Write the equation  in cylindrical coordinates.

Accepted Answer

The answer of Write the equation  in cylindrical coordinates.
...

Question 9

Set up the triple integral  in cylindrical coordinates where Q is the solid above  below  and inside

Accepted Answer

The answer of Set up the triple integral  in...

Question 10

Find the center of mass of the solid with density  and the given shape.

Accepted Answer

The answer of Find the center of mass of the...

Question 11

Compute the volume of the solid bounded by the given surfaces.

Accepted Answer

The answer of Compute the volume of the solid bounded...

Question 12

Rewrite the iterated integral by iterating in the order

Accepted Answer

The answer of Rewrite the iterated integral by iterating in...

Question 13

Evaluate the integral  , where Q is the region with z > 0 bounded by  and  .

Accepted Answer

The answer of Evaluate the integral  , where Q...

Question 14

Numerically estimate the surface area of the portion of  inside of

Accepted Answer

The answer of Numerically estimate the surface area of the...

Question 15

Evaluate the triple integral  .  ,

Accepted Answer

The answer of Evaluate the triple integral  . ...

Question 16

Compute the volume of the solid bounded by the given surfaces.

Accepted Answer

The answer of Compute the volume of the solid bounded...

Question 17

Which if the following could represent the triple integral  in cylindrical coordinates where Q is the region below  and above  ?

Accepted Answer

The answer of Which if the following could represent the...

Question 18

Write the equation  in cylindrical coordinates.

Accepted Answer

The answer of Write the equation  in cylindrical coordinates.
...

Question 19

Which of the following could represent the triple integral  in cylindrical coordinates where Q is the region bounded below by  and above by  ?

Accepted Answer

In cylindrical coordinates, the triple integral is expressed as $\iiint_Q f(r,	heta,z) \ r \ dz \ dr \ d	heta$. Looking at the given options, we can see that only option C has a different value for the $z$-coordinate (compared to the bounds of integration) inside the integral. So, $11eaa948_cd87_30fe_84bc_4f33900d9662_TB2342_11$ represents the triple integral in cylindrical coordinates.

Question 20

Find the mass of the solid with density  and the given shape.  ,

Accepted Answer

The answer of Find the mass of the solid with...

Quiz 14: Multiple Integrals

Find the center of mass of the solid with density and the given shape. ,

Find the mass of the solid with density and the given shape.

Evaluate the integral , where Q is the region with z > 0 bounded by and .

A function is a pdf on the three-dimensional region Q if for all in Q and . Find k such that is a pdf on the region bounded by z = 0, z = 3, x = 0, x = 2, y - z = 0 , and y - z = 3.

Evaluate the triple integral.

Evaluate the triple integral . ,

Evaluate the triple integral . ,

Write the equation in cylindrical coordinates.

Set up the triple integral in cylindrical coordinates where Q is the solid above below and inside

Find the center of mass of the solid with density and the given shape.

Compute the volume of the solid bounded by the given surfaces.

Rewrite the iterated integral by iterating in the order

Evaluate the integral , where Q is the region with z > 0 bounded by and .

Numerically estimate the surface area of the portion of inside of

Evaluate the triple integral . ,

Compute the volume of the solid bounded by the given surfaces.

Which if the following could represent the triple integral in cylindrical coordinates where Q is the region below and above ?

Write the equation in cylindrical coordinates.

Which of the following could represent the triple integral in cylindrical coordinates where Q is the region bounded below by and above by ?

Find the mass of the solid with density and the given shape. ,