Posted by Ted on .
The area is bounded by y= x^3, x= 2 is rotated around the xaxis. Find the volume using calculus.

Calculus 
Reiny,
I will assume that the xaxis was the other boundary of the region we are rotating
Using discs ...
radius = y = x^3
volume = π∫ y^2 dx from 0 to 2
= π∫ x^6 dx from 0 to 2
= π[ (1/7)x^7 ] from 0 to 2
= π((1/7)(2^7)  0 ]
= 128π/7 
Calculus 
Damon,
It is not clear to me how you would do this without using calculus :)
Graph a sketch of it first. The corners are at (0,0) , (2,0) and (2,8)
Lets add up a bunch of thin cylinders with axes along x axis
volume of thin cylinder = pi y^2 dx
so
in from x = 0 to x = 2 of pi (x^3)^2 dx
or pi x^6 dx
or pi (1/7)x^7
or 1/7 (128)
about 18.3 
times pi 
Damon,
I forgot to multiply by pi