Chapter5_Sec2-CompletedNotes(1) - Chapter 5 Section-2 The Definite Integral Definition of a Definite Integral If f is a function defined for a x b we

# Chapter5_Sec2-CompletedNotes(1) - Chapter 5 Section-2 The...

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Chapter 5, Section-2:The Definite IntegralDefinition of a Definite Integral:Iffis a function defined foraxb, we divide theinterval [a, b] intonsubintervals of equal width Δx= (b-a)/n.We letx0(=a), x1, x2, . . . , xn(=b)be the endpoints of these subintervals and we letx*1, x*2, . . . , x*nbe anysample pointsin these subintervals, sox*ilies in theith subinterval [xi-1, xi]. Then thedefinite integral offfromatobisZbaf(x)dx= limn→∞nXi=1f(x*ixprovided that this limit exists. If it does exist, we say thatfisintegrableon [a, b]; that is, the definiteintegralZbaf(x)dxexists.The sumnXi=1f(x*ixis calledRiemann sumafter the German Mathematician Bernhard Riemann(1826-1866).Example 1:Iff(x) =x2-2x, 0x3, evaluate the Riemann sum withn= 6, taking the samplepoints to be right endpoints.
Chapter 5, Sec-2: The Definite IntegralExample 2:The graph ofgis shown below. EstimateZ3-3g(x)dxwith six sub-intervals using (a)right endpoints, (b) left endpoints, and (c) midpoints.