Use a triple integral to find the volume of the solid bounded by the graphs of the equations $z = 2 - y , z = 6 - y ^ { 2 } , x = 0 , x = 3 , y = 0$
A) 22
B) 32
C) 34
D) 26
E) 38

Question

Quizplus · Accepted Answer

The volume of the solid can be found by setting up a triple integral with the given bounds. The bounds for $x$ are from 0 to 3, for $y$ from 0 to the intersection of $z = 2 - y$ and $z = 6 - y^2$, which occurs when $2 - y = 6 - y^2$, solving this gives $y = 2$, and for $z$ from $2 - y$ to $6 - y^2$. The triple integral becomes:$$\int_{0}^{3} \int_{0}^{2} \int_{2-y}^{6-y^2} dz\,dy\,dx$$Evaluating this integral gives the volume of the solid, which is 22 cubic units.

Use a Triple Integral to Find the Volume of the Solid