Question

Evaluate the following integral by interpreting it in terms of areas ∫_(-5)^1(|2+x|-2) dx

Answer 1

\[\int_{-5}^1(|2+x|-2)\, dx\\
\int_{-5}^1|2+x|\, dx - \int_{-5}^12\, dx\]

The 1st integral: break it up into two parts.
\[|x|=\begin{cases} x & , \text{if }x\ge0\\-x & ,\text{if }x<0\end{cases}\]
So:
\[|2+x|=\begin{cases} 2+x & , \text{if }2+x\ge0\\-(2+x) & ,\text{if }2+x<0\end{cases}\]

So \(2+x \ge 0 \implies x\ge-2\)... i.e use 2+x when x is greater than or equal to -2
and \(2+x < 0\implies x<-2, \)... i.e use -(2+x) when x is less than 2

So:
\[\int_{-5}^1|2+x|\, dx - \int_{-5}^12\, dx\\
=\int_{-5}^{-2}-(2+x)\, dx+\int_{-2}^1(2+x)\, dx -\int_{-5}^12\, dx\]

Answer 2

Sorry small typo:
on the line \(2+x<0 \implies\), at the end it should says "i.e. use \(-(2+x)\) when x is less than -2 "

Answer 3

Alternatively, since the function is fairly simple to graph, you can describe the area the integral represents as the area of geometric shapes in the plane:

Notice the shaded region is made up completely of triangles. If you can find their dimensions, you'll be able to find the areas.