LectureNotesMATH150BSec7_2.pdf - Sec 7.2 Volumes MATH 150B...

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Sec 7.2: VolumesMATH 150B: Lecture NotesTopics CoveredPageIntroduction1Introduction+In this section, we are interested in computing the volume of dierent solids of revolution.+The volume of solids are computed following the approach as the computation of areas using Riemannsums. LetSa solid, posed betweenaandbthat we would like to find its volume. We divide the interval[a, b] intonsubintervals [xi-1, xi], withx0=a,xi=x0+Δx, . . . ,xn=b. On each subinterval [xi-1, xi],Sis a ‘slab”Siwith widthΔx=xi-xi-1. We could approximate thei-th slab by a cylinder with baseA(xi) and heightΔx. ThusV(Si) =A(xi)ΔxandVnXi=1A(xi)Δx.The approximation of the volume increases asnincreases, and the exact volume is obtained by taking thelimit asn! 1. Using the definition of integral through the Riemann sums, we obtain:V= limn!1nXi=1A(xi)Δx=ZbaA(x)dx.Example 1Find the volume of a sphere of radiusr.

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