Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Show that an area function is constant with fund. thm of calc

  1. Aug 22, 2012 #1
    1. The problem statement, all variables and given/known data

    I need to show that the area function for a parabola in the first quadrant is constant.

    2. Relevant equations

    [tex] A(a) = \int^a_0 \frac{1}{a}-\frac{x^2}{a^3}\,dx [/tex]

    3. The attempt at a solution

    Computing this integral gives an area of 2/3. Since the area will always be 2/3 and does not depend on the value of a, then the area function is constant. However, my question is about showing that the area function is constant using the Fundamental Theorem of Calculus.

    Can I use the FTC, legitimately? Or is the hypothesis of the FTC not satisfied because of the division by a in the integrand?
    Last edited: Aug 22, 2012
  2. jcsd
  3. Aug 22, 2012 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Another way to show ##A(a)## is constant would be to show ##A'(a)=0##.
  4. Aug 22, 2012 #3
    Right, but wouldn't that be using the FTC? Taking the derivative would result in:

    [tex]= \frac{1}{a}-\frac{a^2}{a^3} = \frac{1}{a}-\frac{1}{a} = 0 [/tex]

    NOTE: I made a mistake in my original area function above. I fixed the typo.
  5. Aug 22, 2012 #4


    Staff: Mentor

    If a = 0, then the area under the parabola between 0 and 0 is clearly zero, so it's reasonable to assume that a > 0. You don't have to consider a < 0, since you're concerned with the area in the first quadrant.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook