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Calculus help?
Find the volume of the solid created by rotating the region between y=x^3, y=8 and x=0 about the y axis.
2 Answers
- lenpol7Lv 712 months ago
The genrral integral is
pi[Int x^3 dy]
We have to convert x^3 to the (x^1/3)^2 = y^2/3 an convert the limits for 'y' which are 0, & 8
Hence
pi (8/0) Inty^(2/3) dy =
.pi [y^(5/3) / (5/3) ] (8/0)
pi [ 3y^(5/3)/ 5 ] (8/0)
pi [3(8^(5/3)5 - 0] =
pi [(3/5)32)]
pi[19.2] =
60.31.... Units^3
- RealProLv 712 months ago
Using cylindrical shells
The region spans from x=0 to x=2 and a strip parallel to the y-axis at x has length 8 - x^3
If the strip is of width dx, then its full rotation about the y-axis will yield a cylindrical shell of volume 2pi x (8-x^3) dx
V = int[0 to 2] 2pi (8-x^3) x dx
Using disks
V = int[0 to 8] pi (y^(1/3))^2 dy
