Question 3
Let $R$ be the region enclosed by the graph of $y=xe^x$, the $x$-axis, and the lines $x=-1$ and $x=1$.
The area of $R$ is closest to
Not quite. 0.74 is $\int_{-1}^{1}xe^x\,dx$ done in one go. The graph is below the $x$-axis from $-1$ to 0, so that piece comes out as $-0.26$ and cancels area. Split at $x=0$ and add the sizes.
That’s it. Step through the hints below to see why.
Not quite. 2.35 is $\int_{-1}^{1}e^x\,dx=e-e^{-1}$. The $x$ in $xe^x$ has gone missing. You also get 2.35 if you use $xe^x$ itself as the antiderivative, split at $x=0$, and subtract the left piece: $\bigl[xe^x\bigr]_{0}^{1}-\bigl[xe^x\bigr]_{-1}^{0}=e-e^{-1}$.
Not quite. 3.09 is $e+e^{-1}$, which is $\bigl[xe^x\bigr]_{-1}^{1}$: the function itself evaluated at the bounds. Splitting at $x=0$ and adding the sizes gives the same $e^{-1}+e$. Either way, that is integrating its derivative, $(x+1)e^x$, not the function.
One integral from $-1$ to 1 gives 0.74, option (A). The piece below the axis comes out negative, so it cancels area instead of adding to it.
Split where the graph crosses the axis, at $x=0$. Both pieces count as positive area, so the total is $0.26+1\approx1.26$.
The single integral subtracts the blue piece from the orange piece, so it comes out smaller than the area.
The area is the size of the blue piece plus the size of the orange piece.
Up to here the graph stays below the axis, so the integral is negative. The area is its size.
Area questions start with the graph. Find where it crosses the $x$-axis.
$$xe^x=0\ \Rightarrow\ x=0$$
$e^x$ is always positive, so the sign of $xe^x$ is the sign of $x$. From $-1$ to 0 the graph is below the axis. From 0 to 1 it is above.
Integrate each piece separately. Technology does this quickly.
$\displaystyle \int_{-1}^{0}xe^x\,dx$$\displaystyle \approx-0.2642$
$\displaystyle \int_{0}^{1}xe^x\,dx$$\displaystyle =1$
The negative answer only tells you that piece is below the axis.
Area is never negative, so take the size of each piece and add.
$\displaystyle \text{Area}\approx0.2642+1$$\displaystyle \approx1.26$
On your calculator you can also do it in one step: $\int_{-1}^{1}\left|xe^x\right|dx\approx1.2642$. The absolute value flips the piece below the axis.
$\displaystyle \int_{-1}^{0}xe^x\,dx$$\displaystyle =-1+2e^{-1}\approx-0.2642$
$\displaystyle \int_{0}^{1}xe^x\,dx$$\displaystyle =1$
$\displaystyle \text{Area}=\left(1-2e^{-1}\right)+1$$\displaystyle =2-2e^{-1}\approx1.26$
The graph crosses the $x$-axis at $x=0$, so the two pieces are integrated separately and their sizes added.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording is from the 2020 QCAA Mathematical Methods external assessment, © State of Queensland (QCAA) 2020, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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