method of shells and volume of a solid

Question

Use the method of shells to find the volume of the solid obtained by revolving the region bounded by y=2x^2 - x^3 and y=0 around the y axis.

Answer 1

Ana,

Imagine a rectangle with area f(x)*(delta x)

Now, imagine moving that rectangle in a circle around a pole--the rectangle is on a string attached to a pole going through a table and you slide the rectangle around the pole in a circle with radius given by the string.

There is a shape create by the rotating rectangle--call it a "shell" and this is a 3-d shape with volume given by the area of a rectangle moving through space along a circle with circumference (2*pi*r)

That is, f(x)*(delta x)*(2pi*radius given by string length )

Now we move to calculus and finding the limit of the sum of all these rotated rectangles where the radius of rotation is the distance from the "rotation pole"--in this case "x"

and we have the integral f(x)*( 2*pi*x)dx

Hope this helps--Andrew F.

Answer 2

Volume V = 2π∫₀²x(2x² - x³)dx = 2π(x⁴/2- x⁵/5)₀² = 2π(16/2 - 32/5) = 16π/5.