103lab8prep.pdf - MA103 Lab Notes Area Under a Curve(Text...

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MA103 Lab NotesArea Under a Curve (Text: 5.1)When the functiony=f(x)0is continuous on[a, b], we can approximate the area between thex-axis andthe graph offover[a, b]as follows:divide[a, b]into apartitionofn1equal subintervals, using thevaluesxi,0in, wherex0=a,xn=bandxi=a+ Δx·i,Δx=b-an;over[xi-1, xi]construct arectangle with heightf(xi)[i.e. using the right endpoint of each subinterval]so that theith rectangle has areaf(xi)·(xi-xi-1) =f(xi)·Δx;add the areas of thenrectangles.This sum is called aRiemann Sum.Riemann Sum:ARn=ni=1f(xi)·Δx,wherexi=a+(b-a)inNote: to use the left endpoints, takexi=a+(b-a)(i-1)nfor 1inin the above formula.If we increase the number of approximating rectangles used, we obtain more accurate approximations to the area.Byallowingnto approach infinity we obtain the exact areaAunderfover [a, b]; that is,A= limn→∞ni=1f(xi)·Δx.One can show that the height of the approximating rectangles can be taken to bef(x*i)wherex*iis any numberin the subinterval [xi-1, xi] – not necessarily the right endpoint – without changing the value ofA.For instance,

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