Math 104: Centroids and Centers of Mass

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1 Mth 104: Centroids nd Centers of Mss Ryn Blir University of Pennsylvni Thursdy Februry 28, 2013 Ryn Blir (U Penn) Mth 104: Centroids nd Centers of Mss Thursdy Februry 28, / 7

2 Outline 1 Applictions of Definite Integrls 2 Center of Mss nd Centroid Ryn Blir (U Penn) Mth 104: Centroids nd Centers of Mss Thursdy Februry 28, / 7

3 Applictions of Definite Integrls Averges The verge of x 1,x 2,...,x n is given by x = Σn i=1 x i n The verge of function f(x) on nd intervl [,b] is given by f = f(x)dx b Ryn Blir (U Penn) Mth 104: Centroids nd Centers of Mss Thursdy Februry 28, / 7

4 Applictions of Definite Integrls Averges The verge of x 1,x 2,...,x n is given by x = Σn i=1 x i n The verge of function f(x) on nd intervl [,b] is given by f = f(x)dx b Exmple: Find the verge of sin(x) on [0,π]. Ryn Blir (U Penn) Mth 104: Centroids nd Centers of Mss Thursdy Februry 28, / 7

5 Center of Mss nd Centroid Center of mss of prticles Let (x 1,y 1 ),(x 2,y 2 ),...(x n,y n ) be prticles in the plne with msses m 1,m 2,...,m n respectively. Then their center of mss is the point (x,y) where x = Σn i=1 x im i Σ n i=1 m i y = Σn i=1 y im i Σ n i=1 m i Ryn Blir (U Penn) Mth 104: Centroids nd Centers of Mss Thursdy Februry 28, / 7

6 Centroid Center of Mss nd Centroid The centroid is the point t which n object constructed of uniform mteril would blnce. This is different thn center of mss. (Intuitive) The centroid of plnr region R = {(x,y) x b,g(x) y f(x)} is the point (x,y) given by the verge vlues of x nd y over R. Ryn Blir (U Penn) Mth 104: Centroids nd Centers of Mss Thursdy Februry 28, / 7

7 Centroid Center of Mss nd Centroid The centroid is the point t which n object constructed of uniform mteril would blnce. This is different thn center of mss. (Intuitive) The centroid of plnr region R = {(x,y) x b,g(x) y f(x)} is the point (x,y) given by the verge vlues of x nd y over R. We cn use symmetry nd intuition to conclude informtion bout the centroid Ryn Blir (U Penn) Mth 104: Centroids nd Centers of Mss Thursdy Februry 28, / 7

8 Centroid Center of Mss nd Centroid The centroid is the point t which n object constructed of uniform mteril would blnce. This is different thn center of mss. (Intuitive) The centroid of plnr region R = {(x,y) x b,g(x) y f(x)} is the point (x,y) given by the verge vlues of x nd y over R. We cn use symmetry nd intuition to conclude informtion bout the centroid Exmple: Find the centroid of the intervl from to b using the notion of integrls s verges Ryn Blir (U Penn) Mth 104: Centroids nd Centers of Mss Thursdy Februry 28, / 7

9 Center of Mss nd Centroid Centroid The centroid of plnr region R = {(x,y) x b,g(x) y f(x)} is the point (x,y) given by, x = x(f(x) g(x))dx f(x) g(x)dx y = 1 2 ((f(x))2 (g(x)) 2 )dx f(x) g(x)dx Ryn Blir (U Penn) Mth 104: Centroids nd Centers of Mss Thursdy Februry 28, / 7

10 Center of Mss nd Centroid Centroid The centroid of plnr region R = {(x,y) x b,g(x) y f(x)} is the point (x,y) given by, x = x(f(x) g(x))dx f(x) g(x)dx y = 1 2 ((f(x))2 (g(x)) 2 )dx f(x) g(x)dx Exmple: Centroid of the portion of the unit disk in the first qudrnt Ryn Blir (U Penn) Mth 104: Centroids nd Centers of Mss Thursdy Februry 28, / 7

11 Center of Mss nd Centroid Centroid The centroid of plnr region R = {(x,y) x b,g(x) y f(x)} is the point (x,y) given by, x = x(f(x) g(x))dx f(x) g(x)dx y = 1 2 ((f(x))2 (g(x)) 2 )dx f(x) g(x)dx Exmple: Centroid of the portion of the unit disk in the first qudrnt Exmple: Centroid of the region between y = 4 x 2 nd the x-xis. Ryn Blir (U Penn) Mth 104: Centroids nd Centers of Mss Thursdy Februry 28, / 7

12 Center of Mss nd Centroid Centroid The centroid of plnr region R = {(x,y) x b,g(x) y f(x)} is the point (x,y) given by, x = x(f(x) g(x))dx f(x) g(x)dx y = 1 2 ((f(x))2 (g(x)) 2 )dx f(x) g(x)dx Exmple: Centroid of the portion of the unit disk in the first qudrnt Exmple: Centroid of the region between y = 4 x 2 nd the x-xis. Exmple: Centroid of the region between y = sin(x) nd y = cos(x) for 0 x π 4. Ryn Blir (U Penn) Mth 104: Centroids nd Centers of Mss Thursdy Februry 28, / 7

13 Center of Mss nd Centroid Center of mss Wht if the mteril mking up R hs vrible density given by ρ(x)? The center of mss of plnr region R = {(x,y) x b,g(x) y f(x)} with density ρ(x) is the point (x,y) given by, x = xρ(x)(f(x) g(x))dx ρ(x)(f(x) g(x))dx y = 1 2 ρ(x)((f(x))2 (g(x)) 2 )dx ρ(x)(f(x) g(x))dx Ryn Blir (U Penn) Mth 104: Centroids nd Centers of Mss Thursdy Februry 28, / 7

Improper Integrals. Type I Improper Integrals How do we evaluate an integral such as

Improper Integrals. Type I Improper Integrals How do we evaluate an integral such as Improper Integrls Two different types of integrls cn qulify s improper. The first type of improper integrl (which we will refer to s Type I) involves evluting n integrl over n infinite region. In the grph