Question

Question: Sketch the solid whose volume is given by the iterated integral. \(\int_{\rm{0}}^{\rm{2}} {\int_{\rm{0}}^{{\rm{2 - y}}} {\int_{\rm{0}}^{{\rm{4 - }}{{\rm{y}}^{\rm{2}}}} {{\rm{dxdzdy}}} } } \).

Short answer

The sketch of solid is drawn.

Step-by-step solution

Define volume

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

Sketching the solid

Describe and sketch the solid generated by the integral, which is limited by,

\({\rm{z = 2 - y}}\),\({\rm{x = 4 - }}{{\rm{y}}^{\rm{2}}}\)

Begin by establishing the boundaries.

\(\begin{array}{c}{\rm{0}} \le {\rm{x}} \le {\rm{4 - }}{{\rm{y}}^{\rm{2}}}\\{\rm{0}} \le {\rm{y}} \le {\rm{2}}\\{\rm{0}} \le {\rm{z}} \le {\rm{2 - y}}\end{array}\)

Determine what we can learn from the integration area.

The solid is in the first octant (i.e., it is confined by the coordinates \({\rm{y = 0,x = 0,z = 0}}\)).

Therefore, solid bounded between planes \({\rm{z = 2 - y}}\) and coordinate planes \({\rm{x = 4 - }}{{\rm{y}}^{\rm{2}}}\).