Area Under √(9 - x²)

Calculus 2 · Axiom Academy

See how geometry and calculus give the same answer for the area under a semicircle. Find the area under the curve from x = -3 to x = 3 — that is, evaluate . We'll do it two ways: with geometry, then with trigonometric substitution. Nice work! You've seen how algebra and geometry converge on the same result. Here's what we learned: Geometric insight: Recognizing as a semicircle lets us use the simple area formula . Algebraic power: The trigonometric substitution turns a hard integral into a manageable one. The connection: confirms both approaches — geometry gives the intuition, calculus gives the rigor. Trig-sub pattern: Whenever you see , reach for to clear the radical with the Pythagorean identity. This is why we study more than one approach in calculus: sometimes geometry gives quick insight, while algebra gives the tools for the cases where geometric intuition isn't obvious.

This is the written version of the interactive lesson above. See the full Calculus 2 course.