Question 1

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

Accepted Answer

D

Question 2

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

Accepted Answer

We can use polar coordinates to describe the circle:

$r = 7$

Then the limits of integration for the double integral are:

$0 \leq 	heta \leq 2\pi$

$0 \leq r \leq 7$

The area of the region R is given by the double integral:

$A = \iint_R dA = \int_{0}^{2\pi} \int_{0}^{7} r dr d	heta = \pi(7^2-0^2) = 49\pi$

Using the hexadecimal converter, we find that the correct answer is B, which corresponds to:

11ea8a3b_b739_7164_9fff_8dfe5cb2ca82_TB6897_11 11ea8a3b_b739_7165_9fff_3fdeb6abc9ed_TB6897_11

Question 3

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

Polar,

Question 4

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

Accepted Answer

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

Question 5

Use polar coordinates to evaluate.

Accepted Answer

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

Question 6

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

Accepted Answer

The volume of a sphere in polar coordinates is given by the formula:
V = (2/3)πr^3

In polar coordinates, the radius of a point is given by:
r = √(x^2 + y^2 + z^2)

For a sphere with radius 11, we can write the equation in polar coordinates as:
r = 11

Substituting this value of r in the formula for volume, we get:

V = (2/3)π(11)^3
  = 11ea8a3b_b738_fc27_9fff_f566e649de39_TB6897_11

Therefore, the correct answer is (C).

Question 7

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

Accepted Answer

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

Question 8

An agricultural sprinkler distributes water in a circular pattern of radius  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  feet centered at the sprinkler?

Accepted Answer

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

Question 9

A cylindrical drill with radius  is used to bore a hole through the center of a sphere of radius  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  is...

Question 10

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

Accepted Answer

The integral is: 
$$\int_{-2}^{2}\int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}}\frac{1}{\sqrt{x^2+y^2+1}}dydx$$
In polar coordinates, $x=r\cos(	heta)$ and $y=r\sin(	heta)$, so we have:
$$\int_{0}^{2\pi}\int_{0}^{2}\frac{r}{\sqrt{r^2+1}}drd	heta$$
Evaluating the integral:
$$\int_{0}^{2\pi}\int_{0}^{2}\frac{r}{\sqrt{r^2+1}}drd	heta=\int_{0}^{2\pi}\bigg[\sqrt{r^2+1}\bigg]_0^2d	heta=4\pi\sqrt{2}-4\pi\approx\boxed{2.50}$$
Therefore, the best choice is (C).

Question 11

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...

Quiz 12: Extension C: Multiple Integrals

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

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

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

Use polar coordinates to evaluate.

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

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

An agricultural sprinkler distributes water in a circular pattern of radius 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 feet centered at the sprinkler?

A cylindrical drill with radius is used to bore a hole through the center of a sphere of radius 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.

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