Question

Define q as the region bounded by the functions u(y)=y32 1 and v(y)=y between y=1 and y=3. if q is rotated around the y-axis, what is the volume of the resulting solid?

Answer 1

To calculate the volume of the solid obtained by rotating the region bounded by the functions u(y)=y³⁻¹ and v(y)=y around the y-axis, use the method of cylindrical shells and integrate the volume formula from y=1 to y=3.

Step-by-step explanation:

To calculate the volume of the solid obtained by rotating the region bounded by the functions u(y)=y³⁻¹ and v(y)=y around the y-axis, we can use the method of cylindrical shells. The volume of each cylindrical shell is given by the formula V = 2πrhΔy, where r is the distance from the y-axis to the shell, h is the height of the shell, and Δy is the thickness of the shell. In this case, the distance from the y-axis to the shell is y, the height of the shell is v(y)-u(y), and the thickness of the shell is dy. Integrating this formula from y=1 to y=3 will give us the total volume of the solid.