find volume using cylindrical shells method

Question

Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about the specified axis.

y = 9x − x², y = 18;  about x = 3

 

Answer 1

For point P(x,y) the radius of the cylindrical shell is 3-x and the thickness of the cylindrical wall is dx. The height of the cylinder is y=9x-x². The volume of the shell is 2π(3-x)(9x-x²)dx, because the volume of a cylindrical shell is 2πrht where r=radius=3-x, h=height=y and t=thickness=dx.

The volume of the solid is the sum of all the shells, which gives us the integral: 2π∫x(3-x)(9-x)dx.

2π∫(27x-12x²+x³)dx for 0≤x≤3=2π[27x²/2-4x³+x⁴/4]₀³.

Inserting the limits we get: 135π/2=212.058 cubic units approx.