When Exact Solutions Fail

Calculus 3 · Axiom Academy

Some integrals and equations have no formula for the answer. Watch how we get the answer anyway — by letting an approximation close in on it. A number that's real, but has no formula Every technique from Calculus 2 — substitution, parts, partial fractions — assumes there's an antiderivative you can write down. But for a function as ordinary as e^ -x^2 , no combination of powers, roots, logs, exponentials, or trig functions is its antiderivative. There is no formula. And yet the area under its curve is a perfectly definite number. Watch the sweep line cross the bell curve y = e^ -x^2 from 0 to 1 . Everything it passes fills in as area underneath, and the running total climbs to a fixed value near 0.7468 . That number is the integral — exact, even though its antiderivative can't be written as a formula. The antiderivative involves the error function , which is defined by this integral — so "solving" it in closed form is circular. The area, though, is just a number the sweep leaves behind. Here's the trick that saves us. Chop the region under e^ -x^2 into thin strips and add up their areas — a Riemann sum. Slide the handle to use more and more strips: the chunky staircase tightens onto the true curve, and your total closes in on the exact 0.7468 . The gap to the real answer shrinks toward zero. This is numerical integration: as (step size ), the sum converges to the exact area. Smarter rules — trapezoid, Simpson's — just close the gap faster.

This is the written version of the interactive lesson above. See the full Calculus 3 course.