The Centroid of a Region The Centroid of a Region Figure 6.4.1 Main Meny o
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Centroid of a Region The Centroid of a Region
The Centroid of a Region Principle 1:Symmetry If the region has an axis of symmetry,then the centroid (x) lies somewhere along that axis.In particular,if the region has a center,then the center is the centroid. Main Meny 5
Main Menu The Centroid of a Region Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Principle 1: Symmetry If the region has an axis of symmetry, then the centroid (x,y) lies somewhere along that axis. In particular, if the region has a center, then the center is the centroid
Principle2:Additivity If the region,having area 4,consists of a finite number of pieces with areas A,..,A and centroids(年，)，（年n,n)，then (6.4.1) xA=x141+…+XnAn and yA=141+…+ynAn. Main Meny C007e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Principle 2: Additivity If the region, having area A, consists of a finite number of pieces with areas A1 , . . . , An and centroids , then ( ) ( ) 1 1 , , , , n n x y x y
The Centroid of a Region Denote the area of by 4. The centroid (,)of can be obtained from the following formulas: XA= xf()dx,A= f(x)P dx. Figure 6.4.2 Main Meny
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Centroid of a Region Denote the area of Ω by A. The centroid of ( ) can be obtained from the following formulas: x y
The Centroid of a Region Example 1 Locate the centroid of the quarter-disk shown in Figure 6.4.4. y= =Vr2-2 ，习 Figure 6.4.4 Main Meny 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Centroid of a Region Example 1 Locate the centroid of the quarter-disk shown in Figure 6.4.4
The Centroid of a Region Example 2 Locate the centroid of the triangular region shown in Figure 6.4.5. y (0,h) y=-ix+h ·区， b 6,0)x Figure 6.4.5 Main Meny C007e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Centroid of a Region Example 2 Locate the centroid of the triangular region shown in Figure 6.4.5
The Centroid of a Region Figure 6.4.6 shows the region between the graphs of two continuous functions f and g.In this case,if has area 4 and centroid (,)then (6.4.3) A=xLf(x)-g(x)]dx.A=(f()P-[g(x)P)dx. Figure 6.4.6 Main Meny o墙8的
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Centroid of a Region Figure 6.4.6 shows the region Ω between the graphs of two continuous functions f and g. In this case, if Ω has area A and centroid , then ( x y, )
The Centroid of a Region Example 3 Locate the centroid of the region shown in Figure 6.4.7. yk (2,4) f(x)=2x 民，可列 g(x)=x2 x Figure 6.4.7 Main Menu cmew59时0
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Centroid of a Region Example 3 Locate the centroid of the region shown in Figure 6.4.7
Pappus's Theorem on Volumes THEOREM 6.4.4 PAPPUS'S THOREM ON VOLUMEST A plane region is revolved about an axis that lies in its plane.If the region does not cross the axis,then the volume of the resulting solid of revolution is the area of the region multiplied by the circumference of the circle described by the centroid of the region: V=2πRA where 4 is the area of the region and R is the distance from the axis ofrevolution to the centroid of the region.(See Figure 6.4.8.) Main Meny C
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Pappus’s Theorem on Volumes
XiS axis centroid Figure 6.4.8 Main Meny c5S的
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved