Question

Find the volume of the solid generated by revolving the region bounded by the lines y and x=0 about:

1) the x-axis
2) the y-axis
3) the line y
4) the line x

Answer 1

To find the volume of the solid generated by revolving the region bounded by the lines y and x=0 about different axes, we can use the method of cylindrical shells.

Step-by-step explanation:

To find the volume of the solid generated by revolving the region bounded by the lines y and x=0 about different axes, we can use the method of cylindrical shells. Let's go through each case:

1. Revolved about the x-axis: To find the volume, we integrate the area of the cross-section multiplied by the height of the solid. The cross-section is a thin cylindrical shell with radius y and height dx. So the volume is given by the integral of the expression 2*pi*y*dx, where y varies from 0 to the upper bound.
2. Revolved about the y-axis: To find the volume, we integrate the area of the cross-section multiplied by the height of the solid. The cross-section is a thin cylindrical shell with radius x and height dy. So the volume is given by the integral of the expression 2*pi*x*dy, where x varies from 0 to the upper bound.
3. Revolved about the line y: To find the volume, we integrate the area of the cross-section multiplied by the height of the solid. The cross-section is a disk with radius x and area pi*x^2. Since the solid is revolved about the line y, the height of each disk is dy. So the volume is given by the expression pi*x^2*dy, where x varies from the lower bound to the upper bound.
4. Revolved about the line x: To find the volume, we integrate the area of the cross-section multiplied by the height of the solid. The cross-section is a disk with radius y and area pi*y^2. Since the solid is revolved about the line x, the height of each disk is dx. So the volume is given by the expression pi*y^2*dx, where y varies from the lower bound to the upper bound.