1120_1 - Week 1 Outline Review the Fundamental Theorem of...

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Week 1 Outline Review the Fundamental Theorem of Calculus Applications of Fundamental Theorem of Calculus The Method of Substitution Compute areas between curves Review trigonometric integrals Questions Why should we want to compute definite integrals? What are they telling us? Answer AREA Explanation If the graph of the function f ( x ) always stays above the x -axis for x between a and b , then the definite integral gives the area under the graph. However, if the graph of f ( x ) dips below the x -axis, the definite integral gives the area above the x -axis minus the area below the x -axis. Fundamental Theorem of Calculus Part I: R b a f ( x ) d x = F ( b ) - F ( a ), where F ( x ) is an antiderivative of f ( x ). Part II: d d x R x a f ( t ) d t = f ( x ) Remark This theorem tells us that integrals and derivatives are inverses of one another. The strange phenomenon here is that the result doesn’t depend in any way on what you put in for the lower limit a .
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