Lecture 23Section 6.4 The Centroid of a Region; Pappus’Theorem on VolumesJiwen HeTest 3• Test 3: Dec. 4-6 in CASA• Material - Through 6.3.Final Exam• Final Exam: Dec. 14-1 7 in CASAYou Might Be Interested to Know ...• I will replace your lowest test score with the percentage grade from thefinal exam (provided it is higher).• I will give an A to anyone who receives 95% or above on the final exam.• I will give a passing grade to anyone who receives at least 70% on the finalexam.Quiz 1What is today?a. Mondayb. Wednesdayc. Fridayd. None of these11 The Centroid of a Region; Pappus’ Theoremon Volumes1.1 The Centroid of a RegionThe Centroid of a RegionThe center of mass of a plate of constant mass density depends only on its s hapeΩ and falls on a point (¯x, ¯y) that is called the centroid.Principle 1: SymmetryIf the region has an axis of symmetry, then the centroid (¯x, ¯y) lies somewherealong that axis. In particular, if the region has a center, then the center is thecentroid.The Centroid of a Region: Principle of AdditivityPrinciple 2: AdditivityIf the region, having area A, consists of a finite number of pieces with areas A1,· · · , Anand centroids (¯x1, ¯y1), · · · , (¯xn, ¯yn), then¯xA = ¯x1A1+ · · · + ¯xnAn,¯yA = ¯y1A1+ · · · + ¯ynAn.2Centroid of a Region below the graph of f (≥ 0)3Let the region Ω under the graph of f have an area A. The centroid (¯x, ¯y) of Ωis given by¯xA =Zbax f(x) dx, ¯yA =Zba12f(x) 2dx.ExampleExample 1. Find the centroid of the quarter-disc shown in the figure below.4ExampleExample 2. Find the centroid of the right triangle shown in the figure below.Centroid of a Region between the graphs of f and g5f(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.ExampleExample 3. Find the centroid of the region shown in the figure below.61.2 Pappus’ Theorem on VolumesPappus’ Theorem on VolumesPappus’ Theorem on VolumesA plane region is revolved about an axis that lies in its plane. If the region doesnot cross the axis, then the volume of the resulting solid 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 the axis to thecentroid of the region.ExampleExample 4. Find the volume of the solids formed by revolving the region, shownin the figure below, (a) about the y-axis, (b) about the y = 5.7ExampleExample 5. Find 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.ExampleExample 6. Find the centroid of the half-discx2+ y2≤ r2, y ≥ 0by appealing to Pappus’s