Question

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

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

Short answer

The volume is$4\ln \left(4\right)+12\ln \left(3\right)cubicunits$

Step-by-step solution

Given information

We are given a rectangular region

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

Evaluate the integral

We have,

$$\begin{gathered} {\int }_R\frac{x}{y}+\frac{y}{x}dA={\int }_1^5{\int }_1^3\frac{x}{y}+\frac{y}{x}dxdy \\ ={\int }_1^5{{[\frac{x^2}{2y}+y\ln x]}^3}_{1}dy \\ ={\int }_1^5(\frac{4}{y}+\ln (3\left)\right)ydy \\ ={{[4\ln y+\ln (3\left)\frac{y^2}{2}\right]}^5}_1 \\ =4\ln \left(4\right)+12\ln \left(3\right) \end{gathered}$$