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SOLUTION The solid E and its projection onto the xy-plane are shown in the figure. The lower and upper surfaces of E are the planes z = 0 and z = x, so we describe E as a type 1 region: ✔2y² ≤x≤ 2,0 ≤zsx) E = {(x, y, z) | -1✔ syst Then, if the density is p(x, y, z)=p, the mass is m = 1 / 1 pov = 1 1² [² -LIE -12² -p =p/" (4-4/²) oy -p 4y 4-4 Myz M-///x LIVE 1-2 sp 20 -3/0 -20 3.2p Because of the symmetry of E and p about the xz-plane, we can immediately say that M = 0 and therefore y0. The other moments are May-///E -LAVE -12² Jo =£ £₂ 2² P - 24p را ارائے eff* 15" (x, y, z)=(. x²dx dy Therefore the center of mass is Myz Mxz Mxy, m m m 18 p dz dx dy ✓dx dy ✓dy 1: dy dy ✓ pov x 1 X dy 0.[ ✓p dz dx dy ✓dx dy pov ✔p dz dx dy 1dx dy X + Vi D!

SOLUTION The solid E and its projection onto the xy-plane are shown in the figure. The lower and upper surfaces of E are the planes z = 0 and z = x, so we describe E as a type 1 region: ✔2y² ≤x≤ 2,0 ≤zsx) E = {(x, y, z) | -1✔ syst Then, if the density is p(x, y, z)=p, the mass is m = 1 / 1 pov = 1 1² [² -LIE -12² -p =p/" (4-4/²) oy -p 4y 4-4 Myz M-///x LIVE 1-2 sp 20 -3/0 -20 3.2p Because of the symmetry of E and p about the xz-plane, we can immediately say that M = 0 and therefore y0. The other moments are May-///E -LAVE -12² Jo =£ £₂ 2² P - 24p را ارائے eff* 15" (x, y, z)=(. x²dx dy Therefore the center of mass is Myz Mxz Mxy, m m m 18 p dz dx dy ✓dx dy ✓dy 1: dy dy ✓ pov x 1 X dy 0.[ ✓p dz dx dy ✓dx dy pov ✔p dz dx dy 1dx dy X + Vi D!

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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EXAMPLE 5 Find the center of mass of a solid of constant density that is bounded by the parabolic cylinder x = 2y² and the planes x = z, z = 0, and x = 2. SOLUTION The solid E and its projection onto the xy-plane are shown in the figure. The lower and upper surfaces of E are the planes z = 0 and z = x, so we describe E as a type 1 region: E = {(x, y, z) | Then, if the density is p(x, y, z)=p, the mass is m = / / / pov = [1 ² [P -/1 =p 4-414 (4-4y¹) dy 4y -4³ Myz = /// [x -LAVE 20 My = ///E -LAVE 4L** L 24p 3.2p Because of the symmetry of E and p about the xz-plane, we can immediately say that Mz = 0 and therefore y = 0. The other moments are ✓ pov x² dx dy (x, y, z)=( =( ✔ sys1✔✔, 2y² ≤x≤ 2,0 sz sx} Therefore the center of mass is Myz Mxz Mxy mm m o dz dx dy ✓dx dy ✓ Bardy ✓dy x l 0, X dy •p dz dx dy ✔ dx dy ✓ pov p dz dx dy 1dx dy + x +² §² 0!
Transcribed Image Text:EXAMPLE 5 Find the center of mass of a solid of constant density that is bounded by the parabolic cylinder x = 2y² and the planes x = z, z = 0, and x = 2. SOLUTION The solid E and its projection onto the xy-plane are shown in the figure. The lower and upper surfaces of E are the planes z = 0 and z = x, so we describe E as a type 1 region: E = {(x, y, z) | Then, if the density is p(x, y, z)=p, the mass is m = / / / pov = [1 ² [P -/1 =p 4-414 (4-4y¹) dy 4y -4³ Myz = /// [x -LAVE 20 My = ///E -LAVE 4L** L 24p 3.2p Because of the symmetry of E and p about the xz-plane, we can immediately say that Mz = 0 and therefore y = 0. The other moments are ✓ pov x² dx dy (x, y, z)=( =( ✔ sys1✔✔, 2y² ≤x≤ 2,0 sz sx} Therefore the center of mass is Myz Mxz Mxy mm m o dz dx dy ✓dx dy ✓ Bardy ✓dy x l 0, X dy •p dz dx dy ✔ dx dy ✓ pov p dz dx dy 1dx dy + x +² §² 0!
My 지 Step 3: Calculation of M_xy мху 11 = = 11 Center of 19 11 - = 2 P f 올 옷 2 S 2P 3 올 3 mass 랗 - 2 (-1) 3 3 } 24 2 ... 2px8 x 6 3 구 32P 구 9 (5) = 이름 등 - (ay2)3) dy (8-8y6) day f3f = - fr2 f(x - Piger) coly 1-848) dy [CR-Bye 투) 옿 6 [8-84 dy x=y2 [8y-왤 = (2 거 n² dn dy 옿 185-응꽃J16 3 8-8] Ly 2 (23) dy old y 169 + t dy 3 0 2 9 28 D dy
Transcribed Image Text:My 지 Step 3: Calculation of M_xy мху 11 = = 11 Center of 19 11 - = 2 P f 올 옷 2 S 2P 3 올 3 mass 랗 - 2 (-1) 3 3 } 24 2 ... 2px8 x 6 3 구 32P 구 9 (5) = 이름 등 - (ay2)3) dy (8-8y6) day f3f = - fr2 f(x - Piger) coly 1-848) dy [CR-Bye 투) 옿 6 [8-84 dy x=y2 [8y-왤 = (2 거 n² dn dy 옿 185-응꽃J16 3 8-8] Ly 2 (23) dy old y 169 + t dy 3 0 2 9 28 D dy
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