Question

Find the volume of the solid formed by revolving the region bounded by the graphs of y = x^2, x = 4, and y = 1 about the y-axis

39pi/2

225pi/2

735pi/2

None of these

I got 39pi/2

Answer 1

using shells,
v = ∫[1,4] 2πrh dx
where r = x and h = y-1 = x^2-1
v = ∫[1,4] 2π(x)(x^2-1) dx = 225π/2

Hmmm. The area I'm rotating is the shaded triangular region shown at

http://www.wolframalpha.com/input/?i=plot+y%3C%3Dx^2,+16%3E%3Dy%3E%3D1,+0%3C%3Dx%3C%3D4

Using discs,
v = ∫[1,16] π(R^2-r^2) dy
where R=4 and r=√y
v = ∫[1,16] π(16-y) dy = 225π/2

You should have known that 39π/2 is way low, due to the Theorem of Pappus, which states that the volume is the area times the length traveled by the centroid. The area is roughly 15*3/2 = 22, and the centroid would be at about x=3, making a circle of circumference 6π, for a guess of 132π

Answer 2

just did this question and 225 pi/2 is the answer. so what steve said