so I have this math problem:
The graphs of $y=f(x)$ and $y=g(x)$ intersect at more than two points. Find the total area of the regions that are bounded above and below by the graphs of $f$ and $g$.
$f(x)=x^{4}-5x^{2}$
$g(x)=-4$
I'm not entirely sure as to where to start. I find the intersections points to be $x=-2, -1, 1, 2$
However, I did this on my calculator when I have to do it by hand.
Well, if you want to find the intersection points by hand, note that if we set $$a(x)=x^2 - 5x + 4$$ then $$f(x)-g(x)=a(x^2).$$ So you just need to factor $a$, which isn't too bad and note that if $x^2$ is a root of $a$, then $x$ is a root of $f(x)-g(x)$ - i.e. an intersection. In general, just notice that there are no odd powers involved in either polynomial, so they are polynomials of $x^2$.
Of course, given that you know the intersection, verifying that they intersect there is rather easy. Note that in $[-2,-1]$, the function $f$ is below $g$, so you integrate $g-f$ and in $[-1,1]$ the function $f$ is above $g$ so you integrate $f-g$ and in $[1,2]$ the function $f$ is again below $g$. It's should be relatively trivial to integrate those polynomials over those integrals.