Q. 62 - Calculus

Question

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

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

Step-by-Step Solution

Answer

Volume of the integral is $e^2-1 cubicunits$

Step 1: Given information

We are given a region

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

Step 2: Evaluate the integral

We have

$$\begin{gathered} {\int }_R\frac{y}{x}{e}^{y}dA={\int }_0^2{\int }_1^e\frac{y}{x}{e}^{y}dxdy \\ ={\int }_0^2{{\left[\ln x\right]}^e}_1·y{e}^{y}dy \\ ={\int }_0^{2}y{e}^{y}dy \\ ={{[y{e}^y-e^y]}^2}_0=e^2-1 \end{gathered}$$