x y Y fjfottrcltxt3yldydx fflatxlytzy.ly l g TIXcm 2 d · Tuesday, February 4, 2020 Math 32B...
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Tuesday, February 4, 2020 Math 32B Worksheet #5 TA: Victoria Kala Problem 1 Center of mass in two dimensions. (a) Find the mass and center of mass of a triangular lamina with vertices (0, 0), (2, 0), and (0, 1) if the density function is δ(x, y)=1+ x +3y. (b) Find the mass and center of mass of a semicircular lamina x 2 + y 2 R 2 ,x ≥ 0 if the density function is δ(x, y)= C p x 2 + y 2 for some constant C . Page 1 Y M ff scxiyldA fjfottrcltxt3yldydx fflatxlytzy.ly odx g y l Ix 1241 1711 Ix 1311 2 5 dX 141 x1 1 2 311 x 14 7 dx M Choi X fo Sz X 18 2 Dx Ex 12 2 44 3102 5Z I I fmass TIXcm a hffpxgcxiyldA g fjft.tt xtx2t3Xy dydx ffffxtkly EXYYly.IT dX f4Cxtx4I Ix 131115 d 3gf2 x 12 2 1 2 1 3 13 11 xttyx Ddx g3 fffsx x2 f xydx 3 gfqx2 SK 34 41 02 3 815 II zp t Il If F't 9cm Tuffy scxiyldA 3 fff.tt tix yt3y7dydX Fff Atx Ety3 y o dx 3 gfoylltxlhzxf ll tzxpjdx g fffzutxkl xttyxy.ci 2 111 47 d fjlI thx 118 414 44 3 11 xttyx2 tzxttf.ly x3 dx 3gfofz zy zx4dx 3 gf3zx 3 yX 14 3 102 843 3 1 3 fcenwofmas.si ZT YngyJ.oEa.roea.RqM ffp8lx.yldA Ijf crjrdrdo cf.IEf od0 cR3ffEdo aIzofI IT ma T 11cm tuff X8lXiy7dA f.IE oRrcosO.C r.rdrdO p fIE2fokr3asOdrdO p3qffI rflf.ocnodo jp.my fIcnod0 3uICsmo1l jz 3 11 till 3 Yan tuff ykxiddA yf f.fi oRrsino.c r.rdrdO p3Tf.ip rY I osinodo f.pl RfIsinodO 3yYfcnO1 j pz 31 10 07 0
Transcript of x y Y fjfottrcltxt3yldydx fflatxlytzy.ly l g TIXcm 2 d · Tuesday, February 4, 2020 Math 32B...
Tuesday, February 4, 2020 Math 32B Worksheet #5 TA: Victoria Kala
Problem 1 Center of mass in two dimensions.
(a) Find the mass and center of mass of a triangular lamina with vertices(0, 0), (2, 0), and (0, 1) if the density function is �(x, y) = 1 + x+ 3y.
(b) Find the mass and center of mass of a semicircular lamina x2 + y2 R2, x � 0 if the density function is �(x, y) = C
px2 + y2 for some constant
C.
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Tuesday, February 4, 2020 Math 32B Worksheet #5 TA: Victoria Kala
Problem 2 Center of mass in three dimensions. Suppose you are eating an ice creamcone. You determine the ice cream cone can be modeled as a solid that liesabove the cone z =
px2 + y2 and below the sphere x2 + y2 + z2 = z. If the
density of the ice cream cone has uniform density �(x, y) = 1, determine themass and center of mass of the ice cream cone.
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Tuesday, February 4, 2020 Math 32B Worksheet #5 TA: Victoria Kala
Problem 3 Probability. The joint density function for a random variables X and Y is
f(x, y) =
(C(x+ y) if 0 x 4, 0 y 3
0 otherwise.
(a) Find the value of the constant C.
(b) Find P (X 3, Y � 1).
(c) Find P (X + Y 1).
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Tuesday, February 4, 2020 Math 32B Worksheet #5 TA: Victoria Kala
Problem 4 Change of Variables.
(a) EvaluateRR
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(b) EvaluateRR
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