Question

Find the volume between the graph of the given function and the xy-plane over the specified rectangle in the xy-plane

$$\begin{gathered} f(x,y)=x^{2}y{e}^{xy} \\ WhereR=\left\{\right(x,y\left)\right|0\le x\le 1,0\le y\le 2\} \end{gathered}$$

Short answer

The integral does not converge

Step-by-step solution

Given information

We are given a rectangular region

$$\begin{gathered} f(x,y)=x^{2}y{e}^{xy} \\ WhereR=\left\{\right(x,y\left)\right|0\le x\le 1,0\le y\le 2\} \end{gathered}$$

Evaluate the integral

We get,

$$\begin{gathered} {\int }_R{x}^{2}y{e}^{xy}dA={\int }_0^2{\int }_0^1{x}^{2}y{e}^{xy}dxdy \\ ={\int }_0^2{{[x^2-\frac{2x}{y}{e}^{xy}]}^1}_{0}dy \\ ={\int }_0^2[1-\frac{2}{y}{e}^y]dy \\ =\infty \end{gathered}$$