Find the volume of the solid formed by revolving the region bounded by the graphs of y = x^3, x = 2, and y = 1 about the y-axis.
A. 93pi/5
B. 120pi/7
C. 47pi/5 <--- My choice
D. None of these
Am I right?
Please help, Thank you!
using shells of thickness dx,
v = ∫[1,2] 2πrh dx
where r=x and h=y-1 = x^3-1
v = ∫[1,2] 2πx(x^3-1) dx = 47π/5
using washers of thickness dy,
v = ∫[1,8] π(R^2-r^2) dy
where R=2 and r=x = ∛y
v = ∫[1,8] π(2^2-y^(2/3)) dy = 47π/5
good work
To find the volume of the solid formed by revolving the region bounded by the graphs of y = x^3, x = 2, and y = 1 about the y-axis, you can use the method of cylindrical shells.
First, you need to find the limits of integration. Since x = 2 is the right boundary of the region, we will integrate from x = 0 to x = 2.
Next, consider a horizontal slice within the region at a distance x from the y-axis. The height of this slice is given by the difference between the upper and lower curves, which is y = x^3 - 1. The circumference of the shell is 2πx, and the thickness of the shell is dx.
The volume of each shell is then given by V_shell = 2πx(y)dx = 2πx(x^3 - 1)dx.
To find the total volume, integrate V_shell from x = 0 to x = 2:
V_total = ∫[0 to 2] (2πx(x^3 - 1))dx
= 2π∫[0 to 2] (x^4 - x)dx
= 2π [ (1/5)x^5 - (1/2)x^2 ] evaluated from x = 0 to x = 2
= 2π [ (1/5)(2^5) - (1/2)(2^2) ]
= 2π [ (1/5)(32) - (1/2)(4) ]
= 2π [ 6.4 - 2 ]
= 2π(4.4)
= 8.8π.
So, the correct answer is not among the given options.