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Math Help - Help with Integral proof

  1. #1
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    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 anti-derivative is even and if a function is even then the anti-derivative that passes through the origin is odd (but not the other anti-derivatives!)).

    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 x-axis. However, I can't seem to piece together what this means so any help would really help me out.
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  2. #2
    Senior Member vincisonfire's Avatar
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    The hint is a really good one.
    Even function  f(a)=f(-a)
    Odd function  -f(a)=f(-a)
    Fundamental theorem of calculus \int_{a}^b f(x) = F(b)-F(a)
    When you integrate an even function then you obtain an odd function.
     \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.
     \int_{-a}^a f_{odd}(x) = F_{even}(a)-F_{even}(-a)=F_{even}(a)-F_{even}(a)=0
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  3. #3
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    Thank you! I managed to get it to work graphically and that helped me confirm my proof via the FTC!
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