Question

let r be the region bounded by the parabola y=x2 and the line y=4 . (a) what is the volume of the solid generated when r is rotated about the line y=4 ?

Answer 1

To find the volume of the solid generated when the region bounded by the parabola y = x^2 and the line y = 4 is rotated about the line y = 4, we can use the method of cylindrical shells.

Step-by-step explanation:

To find the volume of the solid generated when the region bounded by the parabola y = x^2 and the line y = 4 is rotated about the line y = 4, we can use the method of cylindrical shells.

We can divide the region into infinitesimally thin vertical strips, each of width dx. The height of each shell is (4 - x^2), and the radius is the distance between the point (x, x^2) on the parabola and the line y = 4.

Therefore, the volume of each shell is given by dV = 2πx(4 - x^2)dx. By integrating this expression from x = -√4 to x = √4, we can find the total volume of the solid.