Question

Volume by Shells

1. Let R be the region bounded by y = X – – x2 and y = 0. Use the shell method to find the volume of the solid generated when R is revolved about the y- axis.
2. Let R be the region bounded by y = (1 + x2)-2 ,y = 0, x = 0 and x = 1. Use the shell method to find the volume of the solid generated when R is revolved about the y-axis.
3. Let R be the region bounded by y = Vx, y = 0, and x = 4. Use the shell method to find the volume of the solid generated when R is revolved about the x-axis.
4. Let R be the region bounded by y = 4 – x, y = 2 and x = 0. Use the shell method to find the volume of the solid generated when Ris revolved about the x-axis.
5. Let R be the region bounded by y = x, y = 2 - x and y = 0. Use the shell method to find the volume of the solid generated when R is revolved about the x-axis.

Answer 1

To find the volume using the shell method, integrate the volume of each cylindrical shell using the formula V = 2πrhΔy. Determine the range of y-values and set up an integral based on the given curves.

Step-by-step explanation:

To find the volume of the solid generated when the given region is revolved about the y-axis using the shell method, we need to integrate the volume of each cylindrical shell. The formula for the volume of each shell is given as V = 2πrhΔy, where r represents the distance from the axis of rotation to the shell, h represents the height of the shell, and Δy represents the thickness of the shell.

For each question, we can determine the range of y-values by finding the points of intersection between the given curves and the x or y-axis. Then, we can set up an integral to evaluate the volume of each shell by integrating with respect to y.

1. For the first question, the range of y-values is from 0 to 1. The radius of each shell is x, and the height of each shell is given by the difference between the y-values of the two curves. Thus, the integral becomes ∫(2πx)(x - x²)dy from 0 to 1.