I know the integral of a function. Can I find the arc length over an interval without knowing the function?

Question

Can I find the arc-length of a function knowing the integral only?

More specific: is there a way to find the arc-length of an ellipse without using elliptic integrals as we now the area under the first quadrant = πab/4 = integral from 0 to a?

Answer 1

The wording of your question is awkward: ask if you can determine the arc length of a curve knowing only the area under the curve rather than "the integral" since arc length is itself described with integrals.

You ask for an arc length formula that does not use an elliptic integral. The real world is not as nice as you wish it to be: merely because the area has a tidy little formula does not mean the arc length should have a nice formula. And it doesn't: for an ellipse $x^2/a^2 + y^2/b^2 = 1$ that is not a circle, so without loss of generality we can take $0 < b < a$, the arc length integral for the perimeter of the ellipse is (after some changes of variable) $$ 4a\int_0^1 \frac{\sqrt{1 - mt^2}}{\sqrt{1-t^2}}\,dt = 4a\int_0^1 \frac{1-mt^2}{\sqrt{(1 - t^2)(1-mt^2)}}\,dt $$ where $m = 1 - b^2/a^2 \in (0,1)$. Liouville proved in the 1830s that the integrand here does not have an elementary antiderivative. That doesn't mean one particular value of the definite integral (with specific upper bounds, like $0$ and $1$) can't possibly have a simple exact formula not mentioning integrals, comparable to the case of a formula for the total area inside an ellipse. For example, Liouville showed $e^{-t^2}$ has no elementary antiderivative but we can compute $\int_{-\infty}^{\infty} e^{-t^2}\,dt = \sqrt{\pi}$ by other methods. Still, because the integrand for the total arc length of an ellipse does not have an elementary antiderivative, you should not be surprised that nobody has ever found a simple algebraic formula for an ellipse's arc length.

In fact, even the formula $\pi ab$ for the area of the whole ellipse is in a sense not a simple formula: it involves the transcendental number $\pi$, which we are familiar with only because long ago someone gave that number a name and you've learned about in your previous math courses.