Question
Suppose the area under y = -x^2+1 between x = 0 and x = 1 is rotated around the x-axis. Find the volume by using Disk method and shell method.
Answers
Steve
discs of thickness dx:
v = ∫[0,1] πr^2 dx
where r=y = 1-x^2
v = ∫[0,1] π(1-x^2)^2 dx
shells of thickness dy:
v = ∫[0,1] 2πrh dy
where r=y and h=x=√(1-y)
∫[0,1] 2πy√(1-y) dy
To integrate, use
u^2 = 1-y
dy = -2u du
v = ∫[0,1] πr^2 dx
where r=y = 1-x^2
v = ∫[0,1] π(1-x^2)^2 dx
shells of thickness dy:
v = ∫[0,1] 2πrh dy
where r=y and h=x=√(1-y)
∫[0,1] 2πy√(1-y) dy
To integrate, use
u^2 = 1-y
dy = -2u du