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 y-axis.
If you graph the two functions
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:
The other point can be solved by equating y1(x) and y2(x):
cross multiply and switch sides
Take log to solve for x:
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.