Tutorial Note - Week 09 - Solution

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1 Tutoial Note - Week 9 - Solution ouble Integals in Pola Coodinates. a Since + and + 5 ae cicles centeed at oigin with adius and 5, then {,θ 5, θ π } Figue. f, f cos θ, sin θ cos θ sin θ sin θ da sin θ d dθ sin θ d dθ sin θ dθ 69 b {,θ, θ π} Figue 5 f cos θ, sin θ sin sin + da cos cos 6π [ sin ] d dθ cos cos 6 dθ θ 5 Figue Figue θ +. a f, f cos θ, sin θ 6 cosθ sin θ cos θ + sin θ sin θ sin θ sinθ o ejected

2 So, the pola equation sinθ is a cicle centeed at, with adius of fo θ π. {,θ sin θ, θ π} Figue 6. Hence, Volume V f, da sin θ 6 sin θ sin θ cos θ sin θ 6 cosθ sin θ d dθ dθ 5π +.5 θ.5 Figue 6 Note: sin n θ cos θ dθ n + sinn+ θ + C and sin θ cos θ + cos θ. Fo, a sin θ, θ π and b cos θ, π θ π, whee a, b +. b The volume is bounded b the two sufaces and the intesection is { +. So, the intesection is a cicle centeed at oigin with adius of and {,θ, θ π }. The height between the two sufaces is f, 6 f cos θ, sin θ 6. Volume V. a {,, } 6 d dθ dθ 6π is the pat of the unit cicle centeed at the oigin in the fist quadant, so {,θ, θ π }. e + d d e d dθ e dθ π e

3 { b, }, is the uppe pat of the cicle centeed at the oigin with adius of, so {,θ, θ π}. + dd d dθ dθ π Applications of ouble Integals. a m M M ρ, da ρ, da ρ, da dd dd dd d d d Hence, the mass is m and the cente of mass is, ȳ M, M m m,. b {,, and, {, +, } Figue 7 m M M dd dd dd + d 7 + d d 5 Hence, the mass is m 7 and the cente of mass is, ȳ 5,.,, Figue 7

4 . a b I I I I I + + ρ, da ρ, da dd dd + ρ, da I + I 5 dd dd I I + I 97 5 d 9 5 d 5 + d d 69 Tiple Integals. a b. a b Fo dd d + dv e dv [ d d d + [ ] ] d dd d + d e d dd d d 5 d + d d [ ] 6 + d d d 5 e e + d 7 e e d, use integation b-pats and + d d + e dd e d e e + C.

5 c Since is bounded b the plane + and the -plane, so +. is the egion on the -plane and bounded b,, and. {,, } Figue and {,,,, + } dv d d d d 5 + d d Figue d Since is bounded b the plane o 6 and the plane o the -plane, so 6. When the plane ++ 6 intesects the -plane,, the intesection on the -plane is + 6 o. is the pojection of the plane ++ 6 on the -plane and it is a tiangula egion that bounded b,, and. {,, } Figue 9a and {,,,, 6 } Figue 9 dv d d d d 6 d d Figue 9 Figue 9a 5

6 e Since is bounded above b the plane ABC, so we have to find the equation of ABC fist. Like the tangent plane, the equation of a plane involves the nomal vecto. Using the thee points on the plane to fom two vectos and take the coss poduct fo the nomal vecto. AB [,, ] [,, ] and AC [,, ] i j k nomal vecto n [6,, ] So, the equation of the plane is {,, } Figue a and {,,,, } Figue dv d d d d d d C O A Figue. B O B Figue a A f Again, is bounded above b the plane OQ, so we have to find the equation of OQ fist. OQ [,, ], O [,, ] and n [,, ] So, the equation of the plane is +. {,, } Figue a and {,,,, } Figue dv d d d 6 + d 6 + d d 6

7 O.. +. P Figue Q P Q O Figue a Tiple Integals in Clindical Coodinates. a Since is bounded b the plane and, then. Fo the clinde, + cosθ + sin θ since and θ π. {,θ,, θ π, } Figue f,, + f cos θ, sin θ, cos θ + sin θ dv dθ π d d dθ d dθ Figue Figue a b Since is bounded b 9 and the -plane, then 9 o 9. The intesection between 9 and the -plane,, is a cicle on the -plane, i.e., { since and 7

8 θ π. {,θ,, θ π, 9 } Figue f cosθ, sin θ, cos θ + sin θ + dv 9 6 π dθ 5 5 d d dθ 9 d dθ Figue 9 9 Figue a c + cos θ + {,θ,, θ π, cosθ + } f cosθ, sin θ, sin θ dv cos θ+ sin θ sin θ d d dθ + sin θ d dθ 5 sin θ cos θ + d dθ sin θ + sin θ dθ d + since is above the -plane,. {,θ,, θ π, } f cosθ, sin θ, cos θ dv cos θ d d dθ + cos θ dθ π cos θ d dθ

9 . + and 6 6 The intesection fo and 6 is { 6 9 since. Since 6 is on the top of fo, then {,θ,, θ π, 6 } Figue. Volume V dv f,, 6 dθ 6π d d dθ 6 d dθ Figue + 9 Figue a. + The intesection between and a is { a a {,θ, } a, θ π, a f,, K a since. 9

10 mass m M M M a a K d d dθ a K 6 dθ a πk a a K cos θ d d dθ a 5 K cos θ dθ 6 a a a K sin θ d d dθ a K d d dθ Ka dθ a πk a a K a d dθ a K cosθ a d dθ a 5 K sin θ dθ 6 a K 5 M m, ȳ M m, and M m a Theefoe, the mass is m a πk and the cente of mass is,, a. d dθ Tiple Integals in Spheical Coodinates. a + + ρ ρ since ρ B {ρ,θ,φ ρ, θ π, φ π} f,, + + f ρ sin φ cos θ,ρsin φ sin θ,ρcos φ ρ B + + dv ρ ρ sin φ dρdθ dφ sin φ dθ dφ 5 π 5 sin φ dφ π 5 b Since H is the hemispheical egion, then φ π. H { } ρ,θ,φ ρ, θ π, φ π f ρ sin φ cos θ,ρsin φ sin θ,ρcos φ ρ sin φ H + π dv ρ sin φ ρ sin φ dρdθ dφ π π 5 sin φ dθ dφ 5 sin φ dφ π 5 sin φ sin φ cos φ dφ π 5

11 c Since lies in the fist octant, then θ π and φ π. { ρ,θ,φ ρ, θ π, φ } π f ρ sin φ cos θ,ρsin φ sin θ,ρcos φ ρ sin φ sin θ dv ρ sin φ sin θ ρ sin φ dρdθ dφ 5 sin φ sin θ dθ dφ π sin φ dφ π sin φ cos θ dθ dφ d { ρ,θ,φ ρ, θ π, φ π 6} Figue 5 f ρ sin φ cos θ,ρsin φ sin θ,ρcos φ ρ π dv ρ ρ sin φ dρdθ dφ 6 π sin φ dθ dφ 6 π sin φ dφ π ρ φ Figue 5 Figue 5a. { ρ,θ,φ ρ cos φ, θ π, φ π } Figue 6 Volume V dv π f,, cos φ 6 sin φ cos φ dθ dφ ρ sin φ dρdθ dφ π sin φ cos φ dφ π

12 ρ φ Figue 6 Figue 6a. { ρ,θ,φ ρ a, θ π, φ π f ρ sin φ cos θ,ρsin φ sin θ,ρcos φ Kρ cos φ } m M M M a π a Kρ sin φ cos φ dρdθ dφ a πk sin φ dφ a πk dφ π a π a a 5 πk 5 Kρ sin φ cos φ cos θ dρdθ dφ Kρ sin φ cos φ sin θ dρdθ dφ Kρ sin φ cos φ dρdθ dφ sin φ cos φ dφ a5 πk 5 a K sin φ dθ dφ a 5 K 5 sin φ cos φ cos θ dθ dφ a 5 K 5 sin φ cos φ dθ dφ Theefoe, the mass of H is a πk and the cente of mass is, ȳ,,, a 5.