Question

Question: Use a triple integral to find the volume of the given solid. The solid enclosed by the cylinder \({\rm{y = }}{{\rm{x}}^{\rm{2}}}\)and the planes \({\rm{z = 0}}\)and \({\rm{y + z = 1}}\).

Short answer

Volume is \(\frac{{\rm{8}}}{{{\rm{15}}}}\).

Step-by-step solution

Define volume

The quantity of three-dimensional space filled by the matter is referred to as volume.

Evaluating volume

Since the planes \({\rm{y + z = 1}}\) and \({\rm{z = 0}}\) intersect in the \({\rm{xy}}\) plane along the line \({\rm{y = 1}}\),

Region \({\rm{E}}\) can be defined as

\({\rm{\{ (x,y,z)}} \in {\rm{E|0}} \le {\rm{z}} \le {\rm{1 - y,}}{{\rm{x}}^{\rm{2}}} \le {\rm{y}} \le {\rm{1, - 1}} \le {\rm{x}} \le {\rm{1\} }}\)

Then volume is,

Therefore, the volume is \(\frac{{\rm{8}}}{{{\rm{15}}}}\).