Math GRE - #30

What is $$\int_{-3}^3{|x+1|\,dx}?$$

1. 0
2. 5
3. 10
4. 15
5. 20

Solution :

Choice 3 is the answer.

This is a very basic question. The usual method of dealing with absolute values in integrals is to split it into a sum of two integrals and remove the absolute values. Since $|x+1|$ is negative on $[-3, 1)$ and positive on $[-1, 3]\,\,\,$ , we can split the integral into: $$\int_{-3}^3{|x+1|\,dx}=-\int_{-3}^{-1}{(x+1)\,dx}+\int_{-1}^3{(x+1)\,dx}=10.$$ Another way of doing this problem is to imagine the graph of $|x+1|$ and realize that the integral is the sum of two 45-45-90 triangles (one with base 2 and height 2,  the other with base 4 and height 4). Simply add the area of triangles up and we're done.

Bonus question: Given that $c$ is a constant, what is $$\int_{-\infty}^{\infty}{e^{-|x+c|}\,dx}?$$