Which of the following expressions represents the volume of the solid generated when the region bounded by and is rotated about the line ?
A)
B)
C)

Question

Which of the following expressions represents the volume of the solid generated when the region bounded by  and  is rotated about the line  ? ​
A)​ 
B)​ 
C)​

Quizplus · Accepted Answer

When we rotate the region bounded by y = x^2 - 1 and y = x + 1 about the line y = -1, we get a solid with a circular cross section. The radius of this circular cross section is equal to the distance between the line y = x^2 - 1 and y = -1 at each value of x. This distance is equal to x^2 + 2, since the distance between y = x^2 - 1 and y = x + 1 is (x^2 - (x + 1))^2 = (x^2 - x - 1)^2, which simplifies to (x^2 + 2x)^2 = (x^2 + 2)^2. Therefore, the volume of the solid is given by the integral:

V = pi * integral from -1 to 2 of (x^2 + 2)^2 dx

This integral can be evaluated by expanding (x^2 + 2)^2 and using the power rule of integration. The result is:

V = pi * [x^5/5 + 4x^3/3 + 4x] from -1 to 2

V = pi * [(32/5 + 32 + 8/3) - (-2/5 + 4/3 - 4)]

V = pi * (864/15)

V = 172.8/pi

Simplifying this expression gives:

V = 54.943 cubic units

Approximately, V = 54.94.

Therefore, the answer is C.

Which of the Following Expressions Represents the Volume of the Solid