Question

Let r be the region bounded by the following curve. Use the shell method to find the volume of the solid generated when r is revolved about the y-axis: y = x³ - x⁸ + 1, y = 1.

Answer 1

To find the volume of the solid generated by revolving the region bounded by the curve y = x³ - x⁸ + 1 and y = 1 about the y-axis, we can use the shell method. The shell method involves integrating the volumes of cylindrical shells.

Step-by-step explanation:

To find the volume of the solid generated by revolving the region bounded by the curve y = x³ - x⁸ + 1 and y = 1 about the y-axis, we can use the shell method. The shell method involves integrating the volumes of cylindrical shells. Here are the steps:

1. Determine the limits of integration by finding the x-values where the two curves intersect. Set y = x³ - x⁸ + 1 equal to y = 1 and solve for x. The limits of integration are the x-values where the curves intersect.
2. Write the expression for the volume of each cylindrical shell. The radius of each shell is the distance from the y-axis to the curve at that height, which is x. The height of each shell is the difference in y-values between the two curves, which is 1 - (x³ - x⁸ + 1).
3. Integrate the volume expression with respect to x over the limits of integration to find the total volume.