Math 1a: Chapter 5 Review Nicole Ali Section 4.9: Antiderivatives • A function F is called an antiderivative of f on an interval I if )()( xfxF =′for all x in I. • The general antiderivative is CxF+)(, where C is an arbitrary constant. • The graph of the antiderivative F follows the direction field, given by little line segments of slope . )(xf Section 5.1: Areas and Distances • The area A of a region S that lies under the graph of the continuous function f is the limit of the sum of the areas of approximating rectangles: []∑=∞→∞→∞→Δ=Δ++Δ==niinnnnnnxxfxxfxxfRA11)(lim)(...)(limlim • The distance traveled is the area under the graph of the velocity function. • Note that we consider the area under the x-axis as “negative” area. Section 5.2: The Definite Integral • A Riemann Sum is a sum of the form ∑=Δniixxf1*)(• Definition of a Definite Integral. Let f be a continuous function defined for bxa≤≤. Divide the interval [ into n subintervals of width ]ba,nabx /)(−=Δ. Let be the endpoints of those intervals, with nxxx ,...,10ax=0and bxn=. Let be sample points in these subintervals. The definite integral from a to b is . **1,....,nxx∑∫=∞→Δ=niibanxxfdxxf1*)(lim)(• A definite integral gives the net area under a curve • Helpful equations for evaluating definite integrals using Riemann Sums: ∑==nincc1 ∑=+=ninni12)1( ∑∑===niiniiacca11 ∑=++=ninnni126)12)(1( ∑∑==+=+nininiiiiibaba11)(∑=1 ∑=⎥⎦⎤⎢⎣⎡+=ninni1232)1( ∑∑==−=−nininiiiiibaba11)(∑=1 1• The midpoint rule says we can approximate a definite integral by evaluating the Riemann sum at the midpoints of the intervals • Properties of the Definite Integral: ∫∫−=abbadxxfdxxf )()( ∫=aadxxf 0)(∫−=baabccdx )(, where c is any constant ∫∫=babadxxfcdxxcf )()(, where c is any constant ∫∫−=−bababadxxgdxxfdxxgxf )()()]()([∫∫∫∫+=+bababadxxgdxxfdxxgxf )()()]()([ If for , then 0)( ≥xfbxa ≤≤ 0)( ≥∫badxxfIf for )()( xgxf ≥bxa≤≤, then ∫∫≥babadxxgdxxf )()(If for Mxfm ≤≤ )(bxa≤≤, then ∫−≤≤−baabMdxxfabm )()()( Section 5.3: Evaluating Definite Integrals • Evaluation Theorem. If f is continuous on , then , where F is any antiderivative of f. ],[ ba)()()( aFbFdxxfba−=∫• Indefinite Integral. Note that a definite integral gives a number, where an indefinite integral gives a function. means)()( xFdxxf =∫)()( xfxF=′. • Table of Indefinite Integrals: see page 369 of the textbook. Remember, finding indefinite integrals is just like finding antiderivatives. • Net Change Theorem: The integral of a rate of change is the net change: ∫−=baaFbFdxxF )()()(' Section 5.4: The Fundamental Theorem of Calculus • The Fundamental Theorem of Calculus. Suppose f is continuous on . ],[ ba1. If , then ∫=xadttfxg )()()()( xfxg=′. Thus, )()( xfdttfdxdxa=∫. 2. , where ∫−=baaFbFdxxf )()()(fF=′. 2Section 5.5: The Substitution Rule • The Substitution Rule. If )(xgu=is a differentiable function whose range is an interval I, and f is continuous on I, then ∫∫=′duufdxxgxgf )()())((.• The Substitution Rule for Definite Integrals. If g′is continuous on and f is continuous on the range of ],[ ba)(xgu=, then . ∫∫=′)()()()())((bgagbaduufdxxgxgf• Integrals of Symmetric Functions. Suppose f is continuous on ],[ aa−. 1. If f is even [], then . )()( xfxf =−∫∫−=aaadxxfdxxf0)(2)(2. If f is odd [], then . )()( xfxf −=−∫−=aadxxf 0)(
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