Question

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 y-axis.

Answer 1

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 y-axis 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
x¹*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 y-axis, therefore when revolved around the y-axis it makes a cylinder of volume 2πx * (y1(x)-y2(x))dx

Integrate this from 0 to 4 will give the required volume.