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

## Homework Statement

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

## Homework Equations

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

## 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:

LCKurtz
Homework Helper
Gold Member
Another way to show ##A(a)## is constant would be to show ##A'(a)=0##.

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.

Mark44
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.