Question 1

A tank is shaped like an inverted cone with height 9 m and base radius 2 m. What is the correct expression for the area of a horizontal cross-section at height $$y$$ meters above the bottom?

Accepted Answer

$$A(y) = \pi \left(\frac{2}{9}y\$$right)^2$$

Question 2

Given a spherical tank of radius 7 m, what is the area of a horizontal cross-section at height $$y ($$with the center as the origin)?

Accepted Answer

$$A(y) = \pi (49 - y^2$)$$

Question 3

Which of the following integrals correctly represents the work required to pump all the water out of the top of an inverted conical tank (height 9 m, base radius 2 m), using the bottom as the origin?

Accepted Answer

$$\rho g \$$int_0$$^9 (9 - y) \frac{4}{81}\pi y^2$ \, dy$$

Question 4

For a spherical tank of radius 7 m, which integral gives the work to pump all water out through the top (center at origin)?

Accepted Answer

$$\rho g \int_{-7}^7 (7 - y) \pi (49 - y^2$) \, dy$$

Question 5

When using the shell method to find the volume of the solid formed by revolving the region $$R ($$bounded by $$y = 9 - x/2$$, $$y = -1$$, $$y = 2x - 1) $$about $$x = -2$$, what is the radius of a typical shell at position $$x$$?

Accepted Answer

$$x + 2$$

Question 6

For the washer method about $$x = -2$$, what is the outer radius for a washer at height $$y in $$the region $$R$$?

Accepted Answer

$$18 - 2y - (-2)$$

Question 7

Which of the following is the correct shell method integral for the volume when revolving $$R$$ about $$y = -3$$?

Accepted Answer

$$\int_{-1}^7 2\pi (y + 3) [(18 - 2y) - (\frac{y + 1}{2})] \, dy$$

Question 8

When revolving $$R$$ about $$y = -3$$ using the washer method, what is the inner radius for a washer at $$x$$ between $$0$$ and $$4$$?

Accepted Answer

$$(-1) - (-3)$$

Question 9

A conical tank is filled with a liquid of density $$\rho. If $$the work to pump the liquid to the top is $$W = \rho g \$$int_0$$^9 (9 - y) \frac{4}{81}\pi y^2$ \, dy$$, what does the term $$(9 - y)$$ represent?

Accepted Answer

The vertical distance each slice of water must be lifted to reach the top

Question 10

In the context of work required to pump water from a tank, why is the density $$\rho$$ and gravitational acceleration $$g$$ included in the integral?

Accepted Answer

They convert the volume of water moved into the weight, which is necessary for calculating work.

Question 11

Suppose you revolve the region $$R$$ about $$x = -2$$ using the washer method. Why might you need to split the integral into two parts?

Accepted Answer

Because the boundaries of $$R$$ change at $$x = 4$$

Question 12

If the region $$R is $$revolved about $$y = -3$$, what is the geometric meaning of $$(2x - 1) - (-3) in $$the washer method integral?

Accepted Answer

It is the outer radius of the washer at position $$x$$

Question 13

A spherical tank is full of water. If you want to pump the water to a point 2 m above the top of the tank, how would you modify the work integral?

Accepted Answer

Replace $$(7 - y)$$ with $$(9 - y) in $$the integrand

Question 14

Which of the following best describes the shell method for finding the volume of a solid of revolution?

Accepted Answer

It sums the lateral surface areas of cylindrical shells formed by revolving vertical or horizontal slices.

Applications of Integration: Work and Volumes of Revolution

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