AP Calc volume by disks and washers practice questions

A cylindrical water tank has a radius of 3 feet.

What is the volume of water if the tank is 10 feet tall? Use the disk method: Volume = $\displaystyle \frac{1}{3} imes ext{height} imes ext{base area}$ = $\displaystyle 10 imes ext{pi} imes (3)^2 = 90 ext{pi} ext{ cubic feet}$.

True or False: The disk method can be used to find the volume of a solid of revolution.

True. The disk method calculates volume by integrating the area of circular cross-sections perpendicular to the axis of rotation.

Find the volume of revolution of y = x^2 from x = 0 to x = 4.

Using the disk method: $\displaystyle V = ext{pi} imes ext{integral} (x^2)^2 ext{dx}$ from 0 to 4. Result = $\displaystyle \frac{64 ext{pi}}{5}$.

A swimming pool is shaped like a half-cylinder with a diameter of 10 feet.

If it is 30 feet long, find the volume using the disk method: Volume = $\displaystyle \frac{1}{2} imes ext{pi} imes (5)^2 imes 30 = 375 ext{pi} ext{ cubic feet}$.

Complete the sentence: The volume of a solid of revolution is calculated by...

integrating the area of cross-sectional disks along the axis of rotation.

If the radius function is r(x) = 2x, find the volume from x = 1 to x = 3.

Use disk method: $\displaystyle V = ext{pi} imes ext{integral} (2x)^2 ext{dx}$ from 1 to 3. Result = $\displaystyle \frac{32 ext{pi}}{3}$.

True or False: A larger radius results in a smaller volume when using the disk method.

False. A larger radius increases the area of the disks, thus increasing the volume.

A pizza has a radius of 8 inches. What’s the volume of dough?

Assuming a thickness of 1 inch, use: $\displaystyle V = ext{pi} imes (8)^2 imes 1 = 64 ext{pi} ext{ cubic inches}$.

Calculate the volume of a cone with radius 4 feet and height 9 feet.

Using disk method: $\displaystyle V = \frac{1}{3} imes ext{pi} imes (4^2) imes 9 = 48 ext{pi} ext{ cubic feet}$.

Find the volume of revolution of y = 3x + 1 from x = 0 to x = 2.

Use the disk method: $\displaystyle V = ext{pi} imes ext{integral} (3x+1)^2 ext{dx}$ from 0 to 2. Result = $\displaystyle 20 ext{pi}$.

The radius of the disk varies with the equation r(x) = 1 + x.

Find volume from x = 0 to x = 4: $\displaystyle V = ext{pi} imes ext{integral} (1+x)^2 ext{dx}$ from 0 to 4. Result = $\displaystyle \frac{125 ext{pi}}{3}$.

A water fountain has a radius of 2 feet and is 5 feet high. Volume?

Using disk method: $\displaystyle V = ext{pi} imes (2)^2 imes 5 = 20 ext{pi} ext{ cubic feet}$.

True or False: The disk method can only be applied to functions above the x-axis.

False. It can be applied to functions below the x-axis as well.

If the function is f(x) = 4 - x^2, find the volume from -2 to 2.

Using disk method: $\displaystyle V = ext{pi} imes ext{integral} (4-x^2)^2 ext{dx}$ from -2 to 2. Result = $\displaystyle \frac{32 ext{pi}}{3}$.

Calculate the volume of a sphere with a radius of 3 feet.

Using disk method: $\displaystyle V = \frac{4}{3} ext{pi} (3)^3 = 36 ext{pi} ext{ cubic feet}$.

A cylindrical can has a height of 12 inches and a radius of 4 inches. Volume?

Using disk method: $\displaystyle V = ext{pi} imes (4)^2 imes 12 = 192 ext{pi} ext{ cubic inches}$.

A cylindrical tank is 10 ft tall. What is the volume when full?

Volume = $\displaystyle \pi r^2 h$. If radius = 3 ft, then V = $\displaystyle \pi (3)^2 (10) = 90\pi \approx 282.74$ ft³.

Washer method vs Disk method: key difference?

Disk method uses a solid shape, while washer method subtracts volume of inner shape from outer shape.

True or False: Washer method can apply to hollow shapes.

True - It calculates volume by subtracting inner volume from outer volume.

Fill in the blank: The washer method is used when there is a ____ hole.

hollow

Calculate volume using washer method: y = x², y = 4, from x = -2 to x = 2.

Volume = $\displaystyle \pi \int_{-2}^{2} [(4)^2 - (x^2)^2] dx$ = $\displaystyle \pi \int_{-2}^{2} [16 - x^4] dx$.

How to find the outer and inner radii?

Outer radius: distance to outer function. Inner radius: distance to inner function. Example: y = 4, y = x².

A pool shaped like a washer has an outer radius of 8 ft and inner radius of 5 ft. Height is 4 ft. Find volume.

Volume = $\displaystyle \pi \int_{0}^{4} [(8)^2 - (5)^2] dh = \pi \int_{0}^{4} (64 - 25) dh = 39\pi (4) = 156\pi \approx 490.09$ ft³.

True or False: The washer method is only for revolved shapes.

True - It is specifically used for calculating volumes of solids of revolution.

Which scenario uses the washer method? A cylinder with a core or a solid sphere?

A cylinder with a core - the washer method applies when there’s a hollow space.

Calculate the volume of a solid formed by rotating y = x and y = 0 from x = 0 to x = 2.

Use washer method: Volume = $\displaystyle \pi \int_{0}^{2} [(2)^2 - 0^2] dx = \pi \int_{0}^{2} (4) dx = 8\pi$.

A river is bounded by y = 2 and y = x². Find volume from x = -1 to x = 1.

Volume = $\displaystyle \pi \int_{-1}^{1} [(2)^2 - (x^2)^2] dx = \pi \int_{-1}^{1} (4 - x^4) dx$.

Explain the steps to setup a washer problem.

1. Identify outer and inner functions. 2. Determine limits of integration. 3. Set up integral: $\displaystyle V = \pi \int_{a}^{b} [R(y)^2 - r(y)^2] dy$.

A wine barrel has an outer radius of 12 in and inner radius of 8 in. Height is 24 in. Find volume.

Volume = $\displaystyle V = \pi \int_{0}^{24} [(12)^2 - (8)^2] dh = \pi \int_{0}^{24} (144 - 64) dh = 80\pi (24) = 1920\pi \approx 6031.86$ in³.

Washer method setup for y = sin(x) and y = 0 between x = 0 and x = π.

Volume = $\displaystyle \pi \int_{0}^{\pi} [(sin(x))^2 - (0)^2] dx$.

True or False: The area between two curves is always positive.

False - Area can be negative if the lower function is above the upper function in the interval.

Calculate the volume of a washer-shaped fountain.

A fountain has an outer radius of 6 ft and an inner radius of 3 ft. Height is 5 ft. Use the washer method: Volume = π * h * (R² - r²) = π * 5 * (6² - 3²) = π * 5 * (36 - 9) = 135π ft³.