Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.
y=2+1/x^4,y=2,x=4,x=9;
about the x-axis.
Int(2 + 1/x^4 - 2 dx)[4,9]
= -1/3 x^-3 [4,9]
= -1/3(1/729 - 1/64)
= 665/139968
= 0.00475
Sorry - we wanted a solid
v = Int(pi (R^2-r^2) dx)
where R = y = 2+x^-4
and r = 2
pi*Int(2+x^-4)^2 - 4 dx [4,9]
pi*Int(4+4x^-4 + x^-8 - 4 dx)[4,9]
pi*Int(4x^-4 + x^-8 dx) [4,9]
-pi*(28/x^3 + 1/7x^7)
= 0.0597
thank you soo much! :D
Hmm. I didn't see where you specified an axis. I rotated around the x-axis.
To find the volume of the solid obtained by rotating the region bounded by the curves about the x-axis, we can use the method of cylindrical shells.
First, we need to determine the limits of integration for x. The region is bounded by the curves y = 2+1/x^4, y = 2, x = 4, and x = 9. To find the limits of integration for x, we need to determine the values of x where the curves intersect:
- Set y = 2+1/x^4 equal to y = 2:
2+1/x^4 = 2
This simplifies to:
1/x^4 = 0
Since x^4 cannot equal 0, there are no intersection points between the two curves. Therefore, the region is bounded by the curves y = 2+1/x^4 and y = 2, with limits of integration from x = 4 to x = 9.
Next, we need to determine the height of the shells. The height of the shells is equal to the difference in y-values between the curves at a particular x-value. In this case, the height of the shells is given by:
h = (2+1/x^4) - 2
= 1/x^4
Finally, we can calculate the volume of each shell using the formula:
V = 2πrhΔx
Where r is the distance from the axis of rotation to the shell (which is x in this case), h is the height of the shell, and Δx is the thickness of the shell.
Integrating this formula with the given limits of integration, the volume of the solid obtained by rotating the region about the x-axis is:
V = ∫[4, 9] 2πx(1/x^4) dx
Simplifying the integral:
V = 2π ∫[4, 9] (1/x^3) dx
V = 2π [-1/(2x^2)] [4, 9]
V = 2π (-1/162 - (-1/32))
V = 2π (1/32 - 1/162)
V = 2π (130 - 1)/(32*162)
V = 2π (129/5184)
V = π (129/2592)
Therefore, the volume of the solid obtained by rotating the region bounded by the curves y = 2+1/x^4, y = 2, x = 4, and x = 9 about the x-axis is π (129/2592) cubic units.