Ch. 6 – The Definite Integral

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Presentation transcript:

Ch. 6 – The Definite Integral 6.4 – Fundamental Theorem of Calculus

Fundamental Theorem of Calculus Let F(x) represent the antiderivative of f(x). Then… Part I: Part II: Ex: Evaluate: Use the FTC…you don’t even have to find the antiderivative! The lower limit of integration (π above) won’t matter

Ex: Find y’ if . Solve using the FTC again, but the x2 adds a new wrinkle… …CHAIN RULE! For now, let u = x2 , then substitute so that Really, all we did was FTC times the derivative of x2 .

Ex: Find y’ if . Ex: Find y’ if . We don’t want x in the lower limit, so flip the integral! Now we can use FTC and chain rule… Ex: Find y’ if . Since a variable is in both limits of integration, do chain rule on each one separately…

Ex: Find a function y = f(x) with derivative that satisfies f(2) = 7. To get a function for y, use the FTC and integrate both sides! The lower limit of integration is 2 because that’s our known value, but since y(2) = 0, we must add 7!

Ex: Evaluate the following derivatives: