calculus
posted by john .
Find the volume of the solid formed by rotating the region in the 1st quadrant enclosed by the curves y= x^(1/4) and y= x/64 about the yaxis.

If you graph the two functions
y1(x)=x^(1/4) and
y2(x)=x/64
you will see that they meet at two and only two points between which the two curves enclose an area. Rotate this about the yaxis will give the volume.
The two points are:
x=0
y1(0)=0
y2(0)=0
The other point can be solved by equating y1(x) and y2(x):
x^(1/4)=x/64
cross multiply and switch sides
x^{1}*x^(1/4)=64
x^{3/4}=64
Take log to solve for x:
(3/4)log(x)=log(64)
log(x)=(4/3)log(64)
Take antilog to get x=4.
So integration has to be done between 0 and 4.
The elemental vertical slice is of thickness Δx and height (y1(x)y2(x)). The area of each slice is therefore (y1(x)y2(x))Δx.
Each slice is at a distance x from the yaxis, therefore when revolved around the yaxis it makes a cylinder of volume 2πx * (y1(x)y2(x))dx
Integrate this from 0 to 4 will give the required volume.