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

1. Aug 22, 2012

dustbin

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

$$A(a) = \int^a_0 \frac{1}{a}-\frac{x^2}{a^3}\,dx$$

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. Aug 22, 2012

LCKurtz

Another way to show $A(a)$ is constant would be to show $A'(a)=0$.

3. Aug 22, 2012

dustbin

Right, but wouldn't that be using the FTC? Taking the derivative would result in:

$$= \frac{1}{a}-\frac{a^2}{a^3} = \frac{1}{a}-\frac{1}{a} = 0$$

NOTE: I made a mistake in my original area function above. I fixed the typo.

4. Aug 22, 2012

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.