Question

Sketch the region bounded by the curves x=y² and x=y² then use the shell method to find the volume of the solid generated by revolving the region about the y-axis.

Answer 1

To find the volume of the solid generated by revloving a region bounded by curves x=y² and x=y² about the y-axis, we can use the shell method by considering each vertical strip as a cylindrical shell.

Step-by-step explanation:

To sketch the region bounded by the curves x=y² and x=y², we can start by setting them equal to each other to find the points of intersection: y²=y. Solving this equation, we get two solutions: y=0 and y=1. So the region bounded by the curves is a vertical strip between y=0 and y=1.

To find the volume of the solid generated by revolving this region about the y-axis using the shell method, we can consider each vertical strip as a cylindrical shell with thickness dy. The radius of each shell is y², and the height is the difference between the x-values of the curves at each y-value.

Using the formula for volume by the shell method: V = 2π∫(radius)(height)dy, where the integral is taken from y=0 to y=1, we can calculate the volume.