Use the method of cylindrical shells to find the volume V of the solid obtained by rotating the region bounded by the given curves about the x-axis. x = 8y^2 − 2y^3, x = 0

Question

V = ???

Answer 1

x=0 when y=0 and y=4

dV=2π*r(y)*h(y)*dy

r(y)=y

h(y)=-2y³+8y²

V=2π∫y(-2y³+8y²)dy,,,,and the integration is from y=0 to y=4

V=2π[-y⁵/2+2y⁴] evaluated y=0 to y=4

V=2π*102.4=643.4 units cubed