Volume of solid by shell method

Question

Use cylindrical shells to find the volume of the solid formed by rotating the area between the graph of x=y^(7/4) and x=0, 0≤y≤1 about the line y=3. 

Volume = ∫ upper is 1, lower is 0 on integral, f(y)dy where,

f(y) = 

What is the volume?

Answer 1

A cylindrical shell of thickness dy at a position y is caused by

rotating the area of size y^7/4dy around the line y=3.

The radius of the rotation is (3-y).

Therefore the volume of that shell is 2(3-y)πy^7/4dy.

f(y) = 2(3-y)πy^7/4

Lets compute the indefinite integral

F(y) = ∫2(3-y)πy^7/4dy =

6π∫y^7/4dy - 2π∫y^11/4dy =

6π(4/11)y^11/4 - 2π(4/15)y^15/4

The definite integral is F(1)-F(0) =

6π(4/11) - 2π(4/15) = 5.18