Question

Consider the function $f\left(x\right)=4x{e}^{-x^2}$.

(a) Find the signed area of the region between the graph of $f\left(x\right)$and the x-axis on $[-1,2]$shown here:

(b) Find the absolute area of the same region.

Short answer

Part (a) The value is $-2\left[e^{-4}-e^{-1}\right]$.

Part (b) The area is$2\left[e^{-4}-e^{-1}\right]$.

Step-by-step solution

Part (a) Step 1. Given information.

The given function is$f\left(x\right)=4x{e}^{-x^2}$.

Part (a) Step 2. Explanation.

According to the given question,

the integral will be,

$$\begin{gathered} {\int }_{-1}^{2}4x{e}^{-x^2}dx=4{\int }_{-1}^{2}x{e}^{-x^2}dx \\ =-2{\int }_{-1}^{-4}{e}^{u}du \\ =-2{\left[e^w\right]}_{-1}^{-4} \\ =-2\left[e^{-4}-e^{-1}\right] \end{gathered}$$

Part (b) Step 1. Explanation.

The absolute area will be,

$$\begin{gathered} \left|A\right|=\left|-2\left[e^{-4}-e^{-1}\right]\right| \\ =2\left[e^{-4}-e^{-1}\right] \end{gathered}$$