Question

Find the volume of the given solid. Under the surface \(z = xy\)and above the triangle with vertices \((1,1),(4,1),(1,2)\)

Short answer

The volume of the given solid can be:

\(V = \frac{{31}}{8}\).

Step-by-step solution

Find the limits

Consider the surface\(z = xy\)and above the triangular vertices \((1,1),(4,1),(1,2)\)

Find a region D such that volume the surface lies under Z and above the triangular region.

To find the region find the equation of the sides of the triangle with the vertices\((1,1),(4,1),(1,2)\).

Find the equations of the sides by finding the line joining two points.

The equation of the line joining\((1,1)\)and\((4,1)\)is:

\(\begin{aligned}{l}y - {y_0} = m(x - {x_0})\\y - 1 = \frac{{1 - 1}}{{1 - 4}}(x - 1)\\y - 1 = 0(x - 1)\\y - 1 = 0\\y = 1\end{aligned}\)

The equation of the line joining\((1,1)\)and \((1,2)\)is:

\(\begin{aligned}{l}y - 1 = \frac{{2 - 1}}{{1 - 1}}(x - 1)\\y - 1 = 0(x - 1)\\x - 1 = 0\\x = 1\end{aligned}\)

The equation of the line joining\((1,2)\)and\((4,1)\)is:

\(\begin{aligned}{l}y - 1 = \frac{{2 - 1}}{{1 - 4}}(x - 4)\\y - 1 = - \frac{1}{3}(x - 4)\\ - 3(y - 1) = x - 4\\x = 7 - 3y\end{aligned}\)

The region D can be obtained by the following graph.