The Fundamental Theorems of Calculus

The Fundamental Theorems of Calculus Lesson 5.4

First Fundamental Theorem of Calculus • Given f is • continuous on interval [a, b] • F is any function that satisfies F’(x) = f(x) • Then

First Fundamental Theorem of Calculus • The definite integralcan be computed by • finding an antiderivative F on interval [a,b] • evaluating at limits a and b and subtracting • Try

Area Under a Curve • Consider • Area =

Area Under a Curve • Find the area under the following function on the interval [1, 4]

Second Fundamental Theorem of Calculus • Often useful to think of the following form • We can consider this to be a function in terms of x View QuickTime Movie View Geogebra Demo

Second Fundamental Theorem of Calculus • Suppose we aregiven G(x) • What is G’(x)?

Second Fundamental Theorem of Calculus • Note that • Then • What about ? Since this is a constant …

Second Fundamental Theorem of Calculus • Try this

Assignment • Lesson 5.4 • Page 329 • Exercises 1 – 49 odd

The Fundamental Theorems of Calculus