Question

Question: Use a triple integral to find the volume of the given solid. The tetrahedron enclosed by the coordinate planes and the plane \({\rm{2x + y + z = 4}}\).

Short answer

Volume is \(\frac{{{\rm{16}}}}{{\rm{3}}}\).

Step-by-step solution

Define volume

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

Evaluating volume

Consider the given values and simplify,

\({\rm{2x + y + z = 4}}\)

When\({\rm{y = 0}}\)then\({\rm{x = 2}}\),

And when\({\rm{z = 0}}\)then\({\rm{y = 4 - 2x}}\).

The given solid's volume is,

\(\begin{array}{c}{\rm{V = }}\int_{\rm{0}}^{\rm{2}} {\int_{\rm{0}}^{{\rm{4 - 2x}}} {\int_{\rm{0}}^{{\rm{4 - 2x - y}}} {{\rm{dzdydx}}} } } \\{\rm{ = }}\int_{\rm{0}}^{\rm{2}} {\int_{\rm{0}}^{{\rm{4 - 2x}}} {{\rm{(4 - 2x - y)}}} {\rm{dydx}}} \\{\rm{ = }}\int_{\rm{0}}^{\rm{2}} {\left( {{\rm{4y - 2xy - }}\frac{{{{\rm{y}}^{\rm{2}}}}}{{\rm{2}}}} \right)_{{\rm{y = 0}}}^{{\rm{y = 4 - 2x}}}{\rm{dx}}} \\{\rm{ = }}\int_{\rm{0}}^{\rm{2}} {\left( {{\rm{16 - 8x - 8x + 4}}{{\rm{x}}^{\rm{2}}}{\rm{ - }}\frac{{{\rm{16 - 16x + 4}}{{\rm{x}}^{\rm{2}}}}}{{\rm{2}}}} \right)} {\rm{dx}}\\{\rm{ = }}\int_{\rm{0}}^{\rm{2}} {\left( {{\rm{8 - 8x + 2}}{{\rm{x}}^{\rm{2}}}} \right)} {\rm{dx}}\\{\rm{ = }}\frac{{{\rm{16}}}}{{\rm{3}}}\end{array}\)

Therefore, volume is\(\frac{{{\rm{16}}}}{{\rm{3}}}\).