Question

Let r be the region bounded by the curves y=x², y=4, and x=0. Let s be the solid generated by rotating r about the axis x=3. What is the volume of the solid s?

Answer 1

To find the volume of the solid generated by rotating the region bounded by the curves y=x², y=4, and x=0 about the axis x=3, we can use the method of cylindrical shells.

Step-by-step explanation:

To find the volume of the solid generated by rotating the region bounded by the curves y=x², y=4, and x=0 about the axis x=3, we can use the method of cylindrical shells.

The volume of the solid can be calculated by integrating the area of each cylindrical shell along the axis of rotation. Each cylindrical shell has a radius equal to the distance from the axis of rotation (x=3) to the curve y=x², and a height equal to the difference in y-values between the curves y=4 and y=x².

The integral to find the volume of the solid s is given by:

V = 2π∫[(3-x)(4-x²)] dx