Question 1

An electric charge is spread over a rectangular region  Find the total charge on R if the charge density at a point  in R (measured in coulombs per square meter)is 
A)  coulombs
B)  coulombs
C)  coulombs
D)  coulombs

Accepted Answer

To find the total charge on the rectangle, we need to multiply the charge density by the area of the rectangle. The area of the rectangle R is given by:
Area of rectangle R = length x width = (5 m) x (3 m) = 15 m²
So, the total charge on the rectangle R is given by:
Total charge on rectangle R = charge density x Area of rectangle R
Total charge on rectangle R = (2.5 x 10^(-6) C/m²) x (15 m²)
Total charge on rectangle R = 3.75 x 10^(-5) C
Now, we can see that the answer is closest to choice C, which is 11eaae29_f445_0ef4_87fe_cb9ed7d4f3ff_TB2067_11. Therefore, the best choice is C.

Question 2

Use a double integral to find the area of the region R where R is bounded by the circle 
A)  
B)  
C)  
D)

Accepted Answer

The answer of Use a double integral to find the...

Question 3

Find the center of mass of a lamina in the shape of an isosceles right triangle with equal sides of length  if the density at any point is proportional to the square of the distance from the vertex opposite the hypotenuse.Assume the vertex opposite the hypotenuse is located at  ,and that the sides are along the positive axes.
A) 
B) 
C) 
D) 
E) None of these

Accepted Answer

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

Question 4

Evaluate the integral  ,where R is the annular region bounded by the circles  and  by changing to polar coordinates.

Accepted Answer

The answer of Evaluate the integral  ,where R is...

Question 5

A swimming pool is circular with a 60-ft diameter.The depth is constant along east-west lines and increases linearly from 3 ft at the south end to 9 ft at the north end.Find the volume of water in the pool.
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of A swimming pool is circular with a...

Question 6

Use polar coordinates to evaluate.

Accepted Answer

The answer of Use polar coordinates to evaluate. ...

Question 7

Find the volume under  and above the region bounded by  and  .
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Find the volume under  and above...

Question 8

Find the mass of the lamina that occupies the region D and has the given density function.Round your answer to two decimal places.

Accepted Answer

The answer of Find the mass of the lamina that...

Question 9

Find the mass and the center of mass of the lamina occupying the region R,where R is the triangular region with vertices   and  ,and having the mass density 
A)   , 
B)   , 
C)  , 
D)  ,

Accepted Answer

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

Question 10

Use polar coordinates to find the volume of the solid bounded by the paraboloid  and the plane  .
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Use polar coordinates to find the volume...

Question 11

Determine whether to use polar coordinates or rectangular coordinates to evaluate the integral  ,where f is a continuous function.Then write an expression for the (iterated)integral.

Accepted Answer

The answer of Determine whether to use polar coordinates or...

Question 12

Use polar coordinates to find the volume of the solid under the paraboloid  and above the disk  .
A) 
B) 
C) 
D) 
E)

Accepted Answer

The paraboloid can be expressed in polar coordinates as $z=r^2$. The disk is defined by $r\leq \sqrt{3}$ in the $xy$-plane. Therefore, the limits of integration are $0\leq 	heta \leq 2\pi$ and $0\leq r\leq \sqrt{3}$. The volume of the solid is given by the integral:
\begin{align*}
\iiint_V dV &= \int_0^{2\pi} \int_0^{\sqrt{3}} \int_0^{r^2} r\ dz\ dr\ d	heta \
&= \int_0^{2\pi} \int_0^{\sqrt{3}} r^3\ dr\ d	heta \
&= \int_0^{2\pi} \frac{1}{4} (3)^2\ d	heta \
&= \frac{3}{4} (2\pi) \
&= \frac{3\pi}{2}.
\end{align*}
Converting to decimal form, we get the final answer of 11eaae29_f447_31f2_87fe_1f21bdf8edd9_TB2067_11.

Question 13

An agricultural sprinkler distributes water in a circular pattern of radius 100 ft.It supplies water to a depth of  feet per hour at a distance of  feet from the sprinkler.What is the total amount of water supplied per hour to the region inside the circle of radius 15 feet centered at the sprinkler?

Accepted Answer

The answer of An agricultural sprinkler distributes water in a...

Question 14

A cylindrical drill with radius 3 is used to bore a hole through the center of a sphere of radius 9 .Find the volume of the ring-shaped solid that remains.Round the answer to the nearest hundredth.

Accepted Answer

The answer of A cylindrical drill with radius 3 is...

Question 15

Evaluate the iterated integral by converting to polar coordinates.Round the answer to two decimal places.  .
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Evaluate the iterated integral by converting to...

Question 16

A lamina occupies the part of the disk  in the first quadrant.Find its center of mass if the density at any point is proportional to its distance from the x-axis.

Accepted Answer

The answer of A lamina occupies the part of the...

Question 17

Use polar coordinates to find the volume of the sphere of radius 3 .Round to two decimal places.
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Use polar coordinates to find the volume...

Question 18

Sketch the solid whose volume is given by the iterated integral

Accepted Answer

The answer of Sketch the solid whose volume is given...

Question 19

Express the triple integral   as an iterated integral in six different ways using different orders of integration where T is the solid bounded by the planes    and

Accepted Answer

The answer of Express the triple integral   as...

Question 20

Find the moment of inertia about the y-axis for a cube of constant density 3 and side length 6 if one vertex is located at the origin and three edges lie along the coordinate axes.

Accepted Answer

The answer of Find the moment of inertia about the...

Quiz 12: Multiple Integrals

An electric charge is spread over a rectangular region Find the total charge on R if the charge density at a point in R (measured in coulombs per square meter) is

Use a double integral to find the area of the region R where R is bounded by the circle

Evaluate the integral ,where R is the annular region bounded by the circles and by changing to polar coordinates.

A swimming pool is circular with a 60-ft diameter.The depth is constant along east-west lines and increases linearly from 3 ft at the south end to 9 ft at the north end.Find the volume of water in the pool.

Use polar coordinates to evaluate.

Find the volume under and above the region bounded by and .

Find the mass of the lamina that occupies the region D and has the given density function.Round your answer to two decimal places.

Find the mass and the center of mass of the lamina occupying the region R,where R is the triangular region with vertices and ,and having the mass density

Use polar coordinates to find the volume of the solid bounded by the paraboloid and the plane .

Determine whether to use polar coordinates or rectangular coordinates to evaluate the integral ,where f is a continuous function.Then write an expression for the (iterated)integral.

Use polar coordinates to find the volume of the solid under the paraboloid and above the disk .

An agricultural sprinkler distributes water in a circular pattern of radius 100 ft.It supplies water to a depth of feet per hour at a distance of feet from the sprinkler.What is the total amount of water supplied per hour to the region inside the circle of radius 15 feet centered at the sprinkler?

A cylindrical drill with radius 3 is used to bore a hole through the center of a sphere of radius 9 .Find the volume of the ring-shaped solid that remains.Round the answer to the nearest hundredth.

Evaluate the iterated integral by converting to polar coordinates.Round the answer to two decimal places. .

A lamina occupies the part of the disk in the first quadrant.Find its center of mass if the density at any point is proportional to its distance from the x-axis.

Use polar coordinates to find the volume of the sphere of radius 3 .Round to two decimal places.

Sketch the solid whose volume is given by the iterated integral

Express the triple integral as an iterated integral in six different ways using different orders of integration where T is the solid bounded by the planes and

Find the moment of inertia about the y-axis for a cube of constant density 3 and side length 6 if one vertex is located at the origin and three edges lie along the coordinate axes.