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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 b a f ( x)dx F (b) F (a) First Fundamental Theorem of Calculus • The definite integral b a f ( x)dx can be computed by finding an antiderivative F on interval [a,b] evaluating at limits a and b and subtracting • Try 7 3 6x dx Area Under a Curve • Consider y sin x cos x on 0, 2 • Area = 2 0 sin x cos x dx Area Under a Curve • Find the area under the following function on the interval [1, 4] y ( x x 1) x 2 Second Fundamental Theorem of Calculus • Often useful to think of the following form x a f (t )dt • We can consider this to be a function in terms of x View Geogebra Demo x F ( x) f (t )dt a View QuickTime Movie Second Fundamental Theorem of Calculus • Suppose we are given G(x) x G( x) (3t 5)dt 4 • What is G’(x)? Second Fundamental Theorem of Calculus • Note that x F ( x) f (t )dt a Since this is a constant … • Then • What about F ( x) F (a ) d F ( x) F (a ) f ( x) dx a F ( x) f (t )dt x ? Second Fundamental Theorem of Calculus • Try this 1 dt x 1 3t F ( x) 2 dt a F ( x) f (t )dt x F (a ) F ( x) so F '( x) f ( x) Assignment • Lesson 5.4 • Page 329 • Exercises 1 – 49 odd