Shell Method

Question

Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the y-axis.

y=8x-x^2,

y=16,

x=0

Answer 1

Draw the region.  It is bounded below by the parabola y = 8x-x², bounded above by the horizontal line y = 16, and is bounded on the left by the y-axis (the line x = 0).  The rightmost point of the region is     (4, 16).

 

Next, take a thin vertical slice of the region (with width dx) at a distance of x from the y-axis, and revolve the slice about the y-axis to generate a cylindrical shell.  

 

Volume of shell = 2π(radius)(height)(thickness)

 

                      = 2π(x)(16 - (8x-x²))dx = 2πx(16-8x+x²)dx

 

                                                       = 2π(x³-8x²+16x)dx

 

Volume of solid = 2π∫_{(0 to 4)}[x³-8x²+16x]dx

 

                      = 2π[¼x⁴ - (8/3)x³ + 8x²]_{(evaluate from x = 0 to x = 4)}

 

                      = 2π[(64 - 512/3 + 128) - (0)]

 

                      = 128π/3