Answer to Question #221129 in Calculus for Sree

Question

find the volume of the region between the cylinder z=y^2 and the xy plane that is bounded by the plane x=1 x=0 y=-1 y=1

Answer 1

Region projection onto the Oxy plane:

Then

$V = \int {\int\limits_D {z(x,y)dxdy} = \int\limits_0^1 {dx} \int\limits_{ - 1}^1 {{y^2}} } dy = \frac{1}{3}\int\limits_0^1 {\left. {{y^3}} \right|_{ - 1}^1dx} = \frac{1}{3}\int\limits_0^1 {(1 + 1)dx = } \frac{2}{3}\left. x \right|_0^1 = \frac{2}{3}$

Answer: $V=\frac{2}{3}$