Calculus
posted by Beth on .
Find the colume, in cubis units, generated by revolving the region bounded by y=x^3 x=0 and y=8 around the line x=7

Best to use discs here, I think, integrating along y from 0 to 8
V = Int(pi*(R^2  r^2) dy) [0,8]
where r = 7x and R=7 and x = y^1/3
V = Int(pi*(49  (7y^1/3)^2)dy)[0,8]
= pi*Int(49  (49  14y^1/3 + y^2/3) dx)[0,8]
= pi*Int(14y^1/3  y^2/3)[0,8]
= pi*(21/2 y^4/3  3/5 y^5/3)[0,8]
= pi*(21/2 * 16  3/5 * 32)
= pi(168  96/5)
= 148.8pi
= 467.46
What the heck. Let's try shells:
V = Int(2pi*r*h dx)[0,2]
= 2pi Int((7x)*(8x^3) dx)[0,2]
= 2pi Int(x^4  7x^3  8x + 56 dx)[0,2]
= 2pi (1/5 x^5  7/4 x^4  4x^2 + 56x)[0,2]
= 2pi (1/5 * 32  7/4 * 16  4*4 + 112)
= 2pi (74.4)
= 148.8 pi
= 467.46