Calculus
posted by Beth .
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 about the xaxis.
c) Find the volume of the solid obtained by rotating the region bounded by y = 1/x^6, y = 0, x = 3, and x = 9, about the xaxis.
Thanks for your trouble.

I'll do (a)
First, determine where the curves intersect:
6y = y^3 at the point (6√6,√6)
You can do this using discs or shells. Using discs, or washers, you have a stack of washers. The area of a disc of outer radius R and inner hole of radis r is
m(R^2  r^2)
Since 6y > y^3 on the interval desired
The thickness of each disc is dy. So, the volume of the stack of discs is the integral
π Int(R^2  r^2 dy)[0,√6]
= π Int(36y^2  (y^3)^2) dy)[0,√6]
= π Int(36y^2  y^6 dy)[0,√6]
= π (12y^3  1/7 y^7)[0,√6]
= π (12*6√6  1/7 * 216√6)
= 36π√6(2  6/7)
= 36*8π√6/7
Or, if you want to calculate using shells, the volume of a shell is 2πrh
r = x, h = x^(1/3)  x/6
So, the volume of a ring of shells is
2π Int(x(x^1/3  x/6) dx )[0,6^{3/2}]
= 2π Int(x(x^1/3  x/6) dx )[0,6^{3/2}]
= 2π Int(x^{4/3}  x^{2}/6) dx )[0,6^{3/2}]
= 2π(3/7 x^{7/3}  x^{3}/18)[0,6^{3/2}]
= 2π(6^{7/2}  6^{9/2}/18)
= 2π*216√6(3/7  1/3)
= π*216√6(4/21)
= 36*8π√6/7 
Thank you!
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
Find the volume of the solid obtained by rotating the region enclosed by the graphs of y=15x, y=3x+11 and x=0 about the yaxis. 
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^(2x)+1 y=0 x=0 x=0.9 about the yaxis. 
CALCULUS MAJOR HELP!!!!!!
Find the volume of the solid obtained by rotating the region bounded y = 16 x and y = 2 x^2 about y =0 Find the volume of the solid obtained by rotating the region bounded about the xaxis by y=4x^2, x =1, and y = 0 Find the volume … 
Calculus volume stuff
Find the volume of the solid obtained by rotating the region bounded y = 16 x and y = 2 x^2 about y =0 Find the volume of the solid obtained by rotating the region bounded about the xaxis by y=4x^2, x =1, and y = 0 Find the volume … 
Calculus I don't understand
Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. y = 10 x and y = 5 x^2 about y =0 Find the volume of the solid obtained by rotating the region bounded by y=8 x^2, x … 
Calculus
Find the volume of the solid obtained by rotating the region enclosed by the graphs of y=18x, y=3x6 and x=0 about the yaxis V=