Let R be the region bounded by y x2 x 1 and y 0 Use the s

Let R be the region bounded by y = x2, x = 1, and y = 0. Use the shell method to find the volume of the solid generated when R is revoked about the line x = - 10. v = (Type an exact answer in terms of pi.) Find the volume of the solid that is generated when the region in the first quadrant bounded by y = x2, y = 4, and x = 0 is revoked about the following line. x = 2 The volume of the solid generated by revolving the bounded region of the first quadrant about the line x = 2 is . (Type an exact answer, using pi as needed.)

Solution

Use cylindrical shell method.
int 2 x f(x) dx

here f(x)= 4-x^2

and x = 2-x since it is being revolved around x=2

2 int (4-x^2)(2-x) dx from x=0 to x=2

2 int 8-4x -2x^2 +x^3 dx

Antiderivative is

8x -2x^2 -2/3 x^3 +x^4/4

At x=2 we have 16- 8 -16/3 +4

2 *20/3= 40/3