Calculus Help

Question

Find the volume of the solid formed by revolving the region bounded by the graphs of y = x^2, x = 4, and y = 1 about the y-axis.

Answer 1

Draw the region and use the washer method.

 

Rotating a thin horizontal strip of width dy about the y-axis yields a washer with outer radius 4 and inner radius 4-√y.

 

The volume of the resulting washer has volume π[(4)² - (4-√y)²]dy

 

                                                                    = π[16 - 16 + 8√y - y]dy

 

                                                                    = π[8√y - y]dy

 

Volume of solid = π∫_{(from 1 to 16)} [8√y - y]dy

 

                       = π[(16/3)y^3/2 - (1/2)y²]_{(from 1 to 16)}

 

You can evaluate the integral by hand or use a graphing calculator..