find the volume of this solid

Question

How do you find the volume of the region bounded by y=x^2-1 , x=2 , y=0 , is revolved about the y-axis? thank you

Answer 1

The region is bounded above by the curve y = x²-1, is bounded on the right by the vertical line x = 2, and is bounded below by the portion of the x-axis where 1≤x≤2.

 

Using the shell method, the volume of a typical shell is 2π(radius)(height)(thickness)

 

                                                                              = 2πx(x²-1)dx

 

Volume of solid of revolution = 2π∫_{(1 to 2)}[x³-x]dx

 

                                          = 2π(¼x⁴ - ½x²) _{from 1 to 2}

 

                                          = 2π[(4-2) - (¼-½)]

 

                                          = 2π(2.25) = 9π/2