Chapter 6.2, Problem 36E

To determine

The area that is enclosed between the x-axis for $y=x^2{e}^{-x}\text{from}x=-1\text{to}x=3$ .

Answer to Problem 36E

The area is 1.8719.

Explanation of Solution

Given information:

The area enclosed between the x-axis and the graph of $y=x^2{e}^{-x}\text{from}x=-1\text{to}x=3$ .

Formula used:

  ${\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)}dx=\text{NINT}\left(f\left(x\right),x,a,b\right)$

Calculation:

The graph of $y=x^2{e}^{-x}$

  

If $y=f\left(x\right)$ is nonnegative and capable of integration over closed interval $\left[a,b\right]$ , then the area under the curve $y=f\left(x\right)$ from a to b is the integral of f from a to b

So,

  $\text{area}={\displaystyle \underset{-1}{\overset{3}{\int }}\left(x^2{e}^{-x}\right)dx}$

Now, Calculator has a numerical integration capability, which will denote as NINT. In particular, we will use NINT $\left(f\left(x\right)x,a,b\right)$ to denote a calculator approximation of ${\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)}dx$ . Hence,

  ${\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)}dx=\text{NINT}\left(f\left(x\right),x,a,b\right)$

Then understanding that the right-hand side of the equation is an approximation of the left-hand side.

Now, use above expression with this function

  $\begin{array}{l}f\left(x\right)=x^2{e}^{-x}\\ {\displaystyle \underset{-1}{\overset{3}{\int }}\left(x^2{e}^{-x}\right)dx=\text{NINT}}\left(x^2{e}^{-x},x,-1,3\right)\\ =1.8719\end{array}$

Conclusion:

The area is 1.8719.