Question

Find the volume of the solid formed by revolving the region bounded by the graph of y = x³, y = 1, and x = 2 about the line x = 2. This is the green region shown below. Express your answer as a decimal rounded to three places.

a. 3.142
b. 5.236
c. 9.424
d. 12.566

Answer 1

To find the volume of the solid formed by revolving the region bounded by the graph of y = x³, y = 1, and x = 2 about the line x = 2, we can use the method of cylindrical shells.

Step-by-step explanation:

To find the volume of the solid formed by revolving the region bounded by the graph of y = x³, y = 1, and x = 2 about the line x = 2, we can use the method of cylindrical shells.

The formula for the volume of a solid formed by revolving a region about a vertical line is:

V = 2π ∫ (radius) (height) dx

In this case, the radius is 2 - x and the height is x³ - 1.

Integrating from x = 1 to x = 2, we can calculate the volume:

V = 2π ∫ (2 - x)(x³ - 1) dx

After evaluating the integral, the volume is approximately 5.236.