Question

The volume under the surface \({\rm{z}}\left( {{\rm{x}},{\rm{\;y}}} \right) = {\rm{x}} + {\rm{y}}\) and above the triangle in the x-y plane defined by \(\{ 0 \le {\rm{y}} \le {\rm{x\;and\;}}0 \le {\rm{x}} \le 12\)} is _________.

Answer 1

Volume \( = \smallint {\rm{\;}}\smallint {\rm{\;z}}\left( {{\rm{x}},{\rm{\;y}}} \right).{\rm{dxdy}}\)

\( = \mathop \smallint \limits_{{\rm{x}} = 0}^{12} \mathop \smallint \limits_{{\rm{y}} = 0}^{\rm{x}} \left( {{\rm{x}} + {\rm{y}}} \right).{\rm{dydx}}\)

\(\mathop \smallint \limits_{x = 0}^{12} \left( {xy + \frac{{{y^2}}}{2}} \right)_0^xdx = \mathop \smallint \limits_{{\rm{x}} = 0}^{12} \frac{3}{2}{{\rm{x}}^2}\)

\( = \frac{3}{2}.\left( {\frac{{{{\rm{x}}^3}}}{3}} \right)_0^{12} = \frac{{{{\left( {12} \right)}^3}}}{2} = 864\)