I have been trying and trying to solve the following problem (I even used wolframalpha as an extra help, but no success, and I have like 100 calculations in my notebook):
The Task:
Calculate the area that the following graphs form: $$y= \frac{ \left| x \right| }{2}, y= \frac{1}{x^2+1}, y= \frac{1}{4}$$
This is the graph of all three functions:
So, the area I need to calculate is:
What I did: I divided this area in two parts, so we have: $S=S_{1}+ S_{2}$
Then I first calculated $S_{1}$ and after $S_{2}$
$$S_{1}= \int_1^1(-\frac{ \left| x \right| }{2}+\frac{1}{x^2+1})$$
...and here I stopped...
--> I don't know how to integrate $|x|$, when I type that in wolframalpha I get $\frac{1}{2} x^2 sgn(x)+c$. What is that sgn(x)?? Can I calculate without it? (I've never seen or used it).
Thank you in advance!
You have to break up an integral involving $|x|$ into two integrals, one where $x\geq 0$ and use $x$ for $|x|$, and another where $x\leq0$ and use $-x$ for $|x|$.
If you call your functions $f(x) = |x|/2$, $g(x) = 1/(x^2+1)$, and $h(x) = 1/4$, then you want
$\int_{-1}^{-1/2}(g(x)-f(x))dx + \int_{-1/2}^{0}(g(x)-h(x))dx + \int_{0}^{1/2}(g(x)-h(x))dx + \int_{1/2}^{1}(g(x)-f(x))dx$
using the appropriate replacements for $f(x)$ on the two inner intervals as I mentioned.