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.
Respond to this Question
Similar Questions

Calculus
1. Find the volume formed by rotating the region enclosed by x=5y and x=y^3 with y¡Ý0 about the yaxis. 2. Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis y=18x6x^2 … 
calculus edit
1. Find the volume formed by rotating the region enclosed by x=5y and x=y^3 with y is greater than or equal to 0 about the yaxis. 2. Find the volume of the solid obtained by rotating the region bounded by the given curves about the … 
Calculus
This problem set is ridiculously hard. I know how to find the volume of a solid (integrate using the limits of integration), but these questions seem more advanced than usual. Please help and thanks in advance! 1. Find the volume of … 
calculus
Sketch the region bounded by the curves y = x^2, y = x^4. 1) Find the area of the region enclosed by the two curves; 2) Find the volume of the solid obtained by rotating the above region about the xaxis; 3) Find the volume of the … 
Calculus :)
Find the volume of the solid formed by rotating the region enclosed by y=e^(3x)+2, y=0, x=0, x=0.4 about the xaxis. 
Calculus
a) Find the volume formed by rotating the region enclosed by x = 6y and y^3 = x with y greater than, equal to 0 about the yaxis. b) Find the volume of the solid obtained by rotating the region bounded by y = 4x^2, x = 1, and y = 0 … 
calculus
1. Let R be the region in the first quadrant enclosed by the graphs of y=4X , y=3x , and the yaxis. a. Find the area of region R. b. Find the volume of the solid formed by revolving the region R about the xaxis. 
calculus
Find the volume of the solid formed by rotating the region enclosed by y=e^(2x)+1, y=0, x=0,x=0.9 about the yaxis. 
calculus
Find the volume of the solid formed by rotating the region enclosed by y=e^(2x)+1 y=0 x=0 x=0.9 about the yaxis. 
calculus
Find the volume of the solid formed by rotating the region enclosed by y=e^(2x)+1 y=0 x=0 x=0.9 about the yaxis.