U-Substitution for Definite Integrals

Calculus 1 · Axiom Academy

LESSON U-Substitution for Definite Integrals When you substitute, the limits travel too — so you can change them and finish entirely in u , never converting back. 1. The Integral and the Substitution We want the exact area under from x=0 to x=1 . The numerator x is (almost) the derivative of the denominator — a textbook flag for u-substitution. The definite integral we're after The sweep traces the true curve and fills the region we want — the number under the running is the area accumulated so far, heading toward our answer. 2. The Limits Travel With the Substitution Here's the one idea that makes definite u-substitution easy. The bounds 0 and 1 are x -values . Once the integral speaks u , the bounds must too — and you get the new bounds by pushing each old one through the very same rule u=g(x) . The bounds mark where the integral starts and stops. As x runs , u=g(x) runs — same trip, new ruler. With the limits converted, you never need to undo the substitution — you finish in u . For any valid substitution, . The limits don't stay put — they transform from a,b into g(a),g(b) . Rewrite the integral with the new limits and the new integrand , integrate once, and evaluate at u=1 and u=2 . No converting back to x . Watch the area in the u -picture: the region is a different shape than in Step 1, yet it fills to the same total. That equal area is the substitution rule made visible — transforming and its limits leaves the value untouched.

This is the written version of the interactive lesson above. See the full Calculus 1 course.