Volumes by cylindrical shell method

Question

Use the method of cylindrical shells to find the volume  of the solid obtianed by the region bounded by the given curves about the -axis.

  

 

Answer 1

Answer 2

A little awkwardly specified. Are we to rotate the lines for the given equations "around" the x-axis ("around" because (x,y) = (0,0) is already a point of closure of the solid, so x=0 is redundant -- unless I'm missing some essential point?)?

So you could either do this as shells concentric to the x-axis, from y=0 to y=27 -- i.e. within the solid -- OR, take the entire cylindrical region out to x=3, r=27, and subtract shells outside the solid (i.e. in the region y<x^3). it will come to the same result. By the first stratagem:

Each shell has a radius of y, hence a circumference of 2.pi.y , and a length of the value of x at that value of y. Insert these into the summation integral, for y=0 to 27:

int [0,27]( x *2.pi.y dy) = int[0,27]y^(1/3)*2.pi.y dy = int[0,27]2.pi.y^(4/3) dy -- you should be able to take it from there?

-- Cheers, -- Mr. d.