The method of cylindrical shells is a method for finding the volume of a solid of revolution. The idea is to approximate the volume of the solid by a sum of the volumes of cylindrical shells, and then to take the limit as the number of shells goes to infinity.

Here are the steps to solve the problem:

1. Identify the region: The region is bounded by the curves xy = 2, x = 0, y = 2, and y = 4.

2. Sketch the region: It's helpful to draw a sketch of the region to visualize it. You'll see that it's a rectangle with vertices at (0,2), (0,4), (1,2), and (1,4).

3. Set up the integral: The volume V of the solid obtained by rotating the region about the x-axis is given by the integral V = ∫[a,b] 2πrh dx, where r is the radius of the shell (the y-coordinate) and h is the height of the shell (the x-coordinate). In this case, a = 2, b = 4, r = y, and h = 2/y. So we have V = ∫[2,4] 2πy*(2/y) dy = ∫[2,4] 4π dy.

4. Evaluate the integral: ∫[2,4] 4π dy = 4π[y]_2^4 = 4π(4 - 2) = 8π cubic units.

So the volume of the solid is 8π cubic units.

Question

The method of cylindrical shells is a method for finding the volume of a solid of revolution. The idea is to approximate the volume of the solid by a sum of the volumes of cylindrical shells, and then to take the limit as the number of shells goes to infinity.

Here are the steps to solve the problem:

1. Identify the region: The region is bounded by the curves xy = 2, x = 0, y = 2, and y = 4.

2. Sketch the region: It's helpful to draw a sketch of the region to visualize it. You'll see that it's a rectangle with vertices at (0,2), (0,4), (1,2), and (1,4).

3. Set up the integral: The volume V of the solid obtained by rotating the region about the x-axis is given by the integral V = ∫[a,b] 2πrh dx, where r is the radius of the shell (the y-coordinate) and h is the height of the shell (the x-coordinate). In this case, a = 2, b = 4, r = y, and h = 2/y. So we have V = ∫[2,4] 2πy*(2/y) dy = ∫[2,4] 4π dy.

4. Evaluate the integral: ∫[2,4] 4π dy = 4π[y]_2^4 = 4π(4 - 2) = 8π cubic units.

So the volume of the solid is 8π cubic units.

Knowee AI · Accepted Answer

The method of cylindrical shells is a method for finding the volume of a solid of revolution. The idea is to approximate the volume of the solid by a sum of the volumes of cylindrical shells, and then to take the limit as the number of shells goes to infinity.

Here are the steps to solve the problem:

1. Identify the region: The region is bounded by the curves xy = 2, x = 0, y = 2, and y = 4.

2. Sketch the region: It's helpful to draw a sketch of the region to visualize it. You'll see that it's a rectangle with vertices at (0,2), (0,4), (1,2), and (1,4).

3. Set up the integral: The volume V of the solid obtained by rotating the region about the x-axis is given by the integral V = ∫[a,b] 2πrh dx, where r is the radius of the shell (the y-coordinate) and h is the height of the shell (the x-coordinate). In this case, a = 2, b = 4, r = y, and h = 2/y. So we have V = ∫[2,4] 2πy*(2/y) dy = ∫[2,4] 4π dy.

4. Evaluate the integral: ∫[2,4] 4π dy = 4π[y]_2^4 = 4π(4 - 2) = 8π cubic units.

So the volume of the solid is 8π cubic units.

Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.xy = 2, x = 0, y = 2, y = 4

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