## Pages

News: Currently the LaTeX and hidden solutions on this blog do not work on Google Reader.
Email me if you have suggestions on how to improve this blog!

## Wednesday, 31 August 2011

### Math GRE - #30

What is $\int_{-3}^3{|x+1|\,dx}?$
1. 0
2. 5
3. 10
4. 15
5. 20

Solution :

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 $\int_{-\infty}^{\infty}{e^{-|x+c|}\,dx}?$
This webpage is LaTeX enabled. To type in-line formulae, type your stuff between two '\$'. To type centred formulae, type '$' at the beginning of your formula and '$' at the end.