Chapter 9, Problem 55RE

To determine

To find : The area of the region in the first quadrant.

Answer to Problem 55RE

The area is 1 square unit.

Explanation of Solution

Given information :

The curve is $y=x{e}^{-x}$ .

Calculation :

Since, the curve is in first quadrant and the value of x varies from 0 to $\infty$ .

Therefore, area under the curve is given by:

  $\begin{array}{l}A={\displaystyle {\int }_0^{\infty }x{e}^{-x}dx}\\ A=\underset{a\to \infty }{\lim }{\displaystyle {\int }_0^{a}x{e}^{-x}dx}\\ A=\underset{a\to \infty }{\lim }{\left[-x{e}^{-x}\right]}_0^a+\underset{a\to \infty }{\lim }{\displaystyle {\int }_0^a{e}^{-x}dx}\\ A=\underset{a\to \infty }{\lim }{\left[-x{e}^{-x}-e^{-x}\right]}_0^a\\ A=\underset{a\to \infty }{\lim }\left[\left(-a{e}^{-a}-e^{-a}\right)-\left(-0e^{-0}-e^{-0}\right)\right]\\ A=\left[\left(0-0\right)-\left(0-1\right)\right]\\ A=1\end{array}$

Hence,

The area is 1 square unit.