The Centroid of a Region; Pappus' Theorem on VolumesCentroidPappus' TheoremSection 6.4Lecture 23Section 6.4 The Centroid of a Region; Pappus’ Theorem onVolumesJiwen HeDepartment of Mathematics, University of [email protected]/∼jiwenhe/Math1431Jiwen He, University of Houston Math 1431 – Section 24076, Lecture 23 December 4, 2008 1 / 16Section 6.4Test 3Test 3: Dec. 4-6 in CASAMaterial - Through 6.3.Jiwen He, University of Houston Math 1431 – Section 24076, Lecture 23 December 4, 2008 2 / 16Section 6.4Final ExamFinal Exam: D ec. 14-17 in CASAJiwen He, University of Houston Math 1431 – Section 24076, Lecture 23 December 4, 2008 3 / 16Section 6.4You Might Be Interested to Know ...I will replace your lowest test score with the percentage gradefrom the final exam (provided it is higher).I will give an A to anyone who receives 95% or above on thefinal exam.I will give a passing grade to anyone who receives at least70% on the final exam.Jiwen He, University of Houston Math 1431 – Section 24076, Lecture 23 December 4, 2008 4 / 16Section 6.4Quiz 1What is today?a. Mondayb. Wednesdayc. Fridayd. None of theseJiwen He, University of Houston Math 1431 – Section 24076, Lecture 23 December 4, 2008 5 / 16Section 6.4 Centroid Pappus’ TheoremThe Centroid of a RegionThe center of mass of a plate ofconstant mass density dependsonly on its shape Ω and falls on apoint (¯x, ¯y) that is c alled thecentroid.Principle 1: Symm etryIf the region has an axis of symmetry, then the centroid (¯x, ¯y ) liessomewhere along that axis. In particular, if the region has a center,then the center is the centroid.Jiwen He, University of Houston Math 1431 – Section 24076, Lecture 23 December 4, 2008 6 / 16Section 6.4 Centroid Pappus’ TheoremThe Centroid of a Region: Principle of AdditivityPrinciple 2: AdditivityIf the region, having area A, consists of a finite numbe r of pieceswith areas A1, · · · , Anand centroids (¯x1, ¯y1), · · · , (¯xn, ¯yn), then¯xA = ¯x1A1+ · · · + ¯xnAn,¯yA = ¯y1A1+ · · · + ¯ynAn.Jiwen He, University of Houston Math 1431 – Section 24076, Lecture 23 December 4, 2008 7 / 16Section 6.4 Centroid Pappus’ TheoremCentroid of a Region below the graph of f (≥ 0)Let the region Ω under the graph of f have an area A. Thecentroid (¯x, ¯y ) of Ω is given by¯xA =Zbax f (x) dx, ¯yA =Zba12f (x) 2dx.Jiwen He, University of Houston Math 1431 – Section 24076, Lecture 23 December 4, 2008 8 / 16Section 6.4 Centroid Pappus’ TheoremExampleExampleFind the centroid of the quarter-disc shown in the figure below.Jiwen He, University of Houston Math 1431 – Section 24076, Lecture 23 December 4, 2008 9 / 16Section 6.4 Centroid Pappus’ TheoremExampleExampleFind the centroid of the right triangle shown in the figure below.Jiwen He, University of Houston Math 1431 – Section 24076, Lecture 23 December 4, 2008 10 / 16Section 6.4 Centroid Pappus’ TheoremCentroid of a Region between the graphs of f and gf (x) ≥ g (x) ≥ 0 for all x in [a, b].Ω = region between the graphs off (Top) and g (Bottom).Let the region Ω between the graphs of f and g have an area A.The centroid (¯x, ¯y ) of Ω is given by¯xA =Zbaxf (x) − g(x) dx, ¯yA =Zba12f (x) 2−g(x) 2dx.Jiwen He, University of Houston Math 1431 – Section 24076, Lecture 23 December 4, 2008 11 / 16Section 6.4 Centroid Pappus’ TheoremExampleExampleFind the centroid of the region shown in the figure below.Jiwen He, University of Houston Math 1431 – Section 24076, Lecture 23 December 4, 2008 12 / 16Section 6.4 Centroid Pappus’ TheoremPappus’ Theorem on VolumesPappus’ Theorem on VolumesA plane region is revolved about an axis that lies in its plane. Ifthe region does not cross the axis, then the volume of the resultingsolid of revolution isV = 2π¯R A = (area of the region) × (circumference of the circle)where A is the area of the region and¯R is the distance from theaxis to the centroid of the region.Jiwen He, University of Houston Math 1431 – Section 24076, Lecture 23 December 4, 2008 13 / 16Section 6.4 Centroid Pappus’ TheoremExampleExampleFind the volume of the solids formed by revolving the region, shownin the figure below, (a) about the y -axis, (b) about the y = 5.Jiwen He, University of Houston Math 1431 – Section 24076, Lecture 23 December 4, 2008 14 / 16Section 6.4 Centroid Pappus’ TheoremExampleExampleFind the volume of the torus generated by revolving the circulardisc(x − h)2+ (y − k)2≤ c2, h, k ≥ c > 0(a) about the x-axis, (b) about the y -axis.Jiwen He, University of Houston Math 1431 – Section 24076, Lecture 23 December 4, 2008 15 / 16Section 6.4 Centroid Pappus’ TheoremExampleExampleFind the centroid of the half-discx2+ y2≤ r2, y ≥ 0by appealing to Pappus’s theorem.Jiwen He, University of Houston Math 1431 – Section 24076, Lecture 23 December 4, 2008 16 /