Question

Question: Sketch the solid whose volume is given by the iterated integral. \(\int_{\rm{0}}^{\rm{1}} {\int_{\rm{0}}^{{\rm{1 - x}}} {\int_{\rm{0}}^{{\rm{2 - 2z}}} {{\rm{dydzdx}}} } } \).

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

\({\rm{V = \{ (x,y,z):0}} \le {\rm{x}} \le {\rm{1,0}} \le {\rm{z}} \le {\rm{1 - x,0}} \le {\rm{y}} \le {\rm{2 - 2z\} }}\)

Plane \({\rm{y = 2 - 2z}}\)

\({\rm{xz}}\)-plane projection

\({\rm{x = 0}}\)and\({\rm{x = 1}}\)

\({\rm{z = 0}}\) and \({\rm{z = 1 - x}}\)

Finished sketch.

The \({\rm{y}}\)-axis is vertical.

Therefore, sketch inside.