
Help with Integral proof
I'm confused about this question so some help would be appreciated.
"Suppose f is an even function and g is an odd function. Explain for any a>0, Integral (from a to a) f =2 Integral (from 0 to a) f and Integral (from a to a) g=0.
First make a sketch of such functions and look at net areas. Can you also explain these results with the Fundamental Theorem of Calculus?
(hint: if a function is even (or odd) then its derivative is odd (or even); and if a function is
odd then any antiderivative is even and if a function is even then the antiderivative that passes through the origin is odd (but not the other antiderivatives!)).
I have no clue how to proceed with this question. I tried graphing even and odd functions. I graphed y=x^2 as well as y=x^4 and I noticed that the graphs had the same magnitude of area from the origin to a and a, respectively. Odd functions, I used y=x^3 and x^5 and these graphs had the same area from a to 0 and 0 to a but opposite sides of the xaxis. However, I can't seem to piece together what this means so any help would really help me out.

The hint is a really good one.
Even function $\displaystyle f(a)=f(a) $
Odd function $\displaystyle f(a)=f(a) $
Fundamental theorem of calculus $\displaystyle \int_{a}^b f(x) = F(b)F(a) $
When you integrate an even function then you obtain an odd function.
$\displaystyle \int_{a}^a f_{even}(x) = F_{odd}(a)F_{odd}(a)=F_{odd}(a)+F_{odd}(a)=2F_{odd}(a) $
When you integrate an odd function then you obtain an even function.
$\displaystyle \int_{a}^a f_{odd}(x) = F_{even}(a)F_{even}(a)=F_{even}(a)F_{even}(a)=0 $

Thank you! I managed to get it to work graphically and that helped me confirm my proof via the FTC!