Question

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

$$\begin{gathered} f(x,y)=xy \\ WhereR=\left\{\right(x,y\left)\right|-2\le x\le 3,-1\le y\le 5\} \end{gathered}$$

Short answer

The value of integral is 24cubicunits

Step-by-step solution

Given information

We are given an region

$$\begin{gathered} f(x,y)=xy \\ WhereR=\left\{\right(x,y\left)\right|-2\le x\le 3,-1\le y\le 5\} \end{gathered}$$

Find the volume

$$\begin{gathered} f(x,y)>0when{R}_1=\left\{\right(x,y\left)\right|0\le x\le 3,0\le y\le 5\left\} \\ and \\ f\right(x,y)<0when{R}_1=\{(x,y)|-2\le x\le 0,-1\le y\le 0\} \end{gathered}$$

And

$$\begin{gathered} {\int }_{R}xydA={\int }_{R_1}xydA-{\int }_{R_2}xydA \\ ={\int }_0^5{\int }_0^{2}xydxdy-{\int }_{-1}^0{\int }_{-2}^{0}xydxdy \\ =\left[\frac{x^2{y}^2}{4}\right]-\left[\frac{x^2{y}^2}{4}\right] \\ =25-1 \\ =24cubicunits \end{gathered}$$