Calculus
posted by Beth on .
Revolve the region bounded by y=x and y=x^2 about the y axis. In cubic units, the resulting volume is?

You can use discs, integrating along y:
V = Int(pi (R^2  r^2) dy)[0,1]
where R = y and r = sqrt(y)
= pi*Int(y  y^2)dy[0,1]
= pi(1/2 y^2  1/3 y^3)[0,1]
= pi(1/2  1/3)
= pi/6
Or, you can use shells, integrating along x:
V = Int(2pi*r*h dx)[0,1]
where r = x h = xx^2
= 2pi*Int(x(xx^2) dx)[0,1]
= 2pi(x^2  x^3 dx)[0,1]
= 2pi(1/3 x^3  1/4 x^4)[0,1]
= 2pi(1/3  1/4)
= 2pi(1/12)
= pi/6 
Revolve the region bounded by y = 4x and y = x2 about the yaxis. In cubic units, the resulting volume is