Quick Calculus Problem! Please Help!! Thank you!

Question

Find the volume of the solid formed by revolving the region bounded by the graphs of y = x2, x = 3, and y = 4 about the y-axis.

a.  (25π / 2)

 

b. (97π / 3)

 

c. (173π / 3)

 

d. None of these

Answer 1

Draw a picture:

                   (3,9)

                    /|

                  /  |

       y=x² /     |  x=3

            /        |

       /             |

(2,4)------------ (3,4)

          y=4

 

 

Shells:

 

At any x from 2 to 3, y ranges from 4 to x².

 

The shell has volume dV = 2πxh dx = 2πx(x² - 4) dx = 2π(x³ - 4x) dx

 

So the volume is:

 

V = ∫₂³ 2π(x³ - 4x) dx = 2π(x⁴/4 - 2x²) |₂³

 

 = 2π(81/4 - 18) - 2π(4 - 8) = 25π/2

 

This means it is choice a