Math 9 Assignment 9 olutions 1. Evaluate 4y + 1 d whee is the fist octant pat of y x cut out by x + y + z 1. olution We need a paametic epesentation of the suface. (x, z). Now detemine the nomal vecto: x 1, x,, z,, 1, In vecto fom this is: N x z x, 1,. x, x, z fo To detemine the domain, we find the cuve of intesection of the suface : y x and the plane: x + y + x 1: } y x z 1 x x, x + y + z 1 then, fo the pat in the fist octant: } z 1 x x x z + x 1 x 1 +. Thus, the domain is given by {(x, z) R ; x 1 +, z 1 x x }. Now to evaluate the suface integal: 4y + 1 d 4x + 1 N dz dx 1+ 1 x x (4x + 1) dz dx 1 5 ( 61 + 44 ).. Evaluate xy d whee is the fist octant pat of z x + y cut out by x + y 1. olution We need a paametic epesentation of the suface. ince is a suface of evolution we can use pola coodinates, so in vecto fom this is: t cos θ, t sin θ, t fo (t, θ), whee we have used t in place of to avoid confusion with the position vecto. Now detemine the nomal vecto: t cos θ, sin θ, 1, t sin θ, t cos θ,, N t t cos θ, t sin θ, t. The domain consists of the potion of the suface : z x + y that lies within the fist octant and within the cylinde x + y 1. In pola coodinates (t, θ), this gives {(t, θ) R ; t 1, θ π/}. Now to evaluate the suface integal: xy d t sin θ cos θ N dt dθ π/ 1 t 3 sin θ cos θ dt dθ 8. 1
Math 9 Assignment 9 olutions 3. Calculate the suface aea of the cuved potion of a ight cicula cone of adius R and height h. olution We need a paametic epesentation of the suface. ince is a suface of evolution we can use pola coodinates, so in vecto fom this is: t cos θ, t sin θ, ht R fo (t, θ), whee we have used t in place of to avoid confusion with the position vecto. Now detemine the nomal vecto: t cos θ, sin θ, h, R t sin θ, t cos θ,, N t ht ht cos θ, sin θ, t. R R The domain consists of the potion of the suface : z (h/r) x + y that lies within the cylinde x + y R. In pola coodinates (t, θ), this gives {(t, θ) R ; t R, θ π}. Now to evaluate the suface integal: d N dt dθ π R t 1 + (h /R ) dt dθ πr R + h. 4. Evaluate and z R. d x + y whee is the pat of the sphee x + y + z 4R between the planes z olution We need a paametic epesentation of the sphee. The easiest way to paemeteize the sphee is to use spheical coodinates with ρ R, so in vecto fom this is: R sin ϕ cos θ, sin ϕ sin θ, cos ϕ fo (t, θ). Now detemine the nomal vecto: ϕ R cos ϕ cos θ, cos ϕ sin θ, sin ϕ, N t 4R sin ϕ cos θ, sin ϕ sin θ, sin ϕ cos ϕ. R sin ϕ sin θ, sin ϕ cos θ,, The domain consists of the potion of the suface that lies within the planes z and z R. In spheical coodinates these coespond to ϕ π/ and ϕ π/3 espectively. Thus, the domain is: {(ϕ, θ) R ; π/3 ϕ π/, θ π}. Now to evaluate the suface integal: d x + y N 4R sin dϕ dθ ϕ π π/ π/3 csc ϕ dϕ dθ π ln 3.
Math 9 Assignment 9 olutions 3 5. Evaluate (yz i + ye x j + x k ) n d whee is defined by y x, y 4, z 1, and n is the unit nomal to the suface with positive y-component. olution We need a paametic epesentation of the suface. In vecto fom this is: x, x, z fo (x, z), whee {(x, z) R ; x, z 1}. Now detemine the nomal vecto: x 1, x,, z,, 1, Now, to evaluate the suface integal: (yz i + ye x j + x k ) n d 1 N x z x, 1,. (x z i + x e x j + x k ) N dx dz ( x 3 z + x e x ) dx dz e 1e. 6. Evaluate (x i + y j ) n d whee is the pat of z x + y below z 1, and n is the unit nomal to the suface with negative z-component. olution We need a paametic epesentation of the suface. ince is a suface of evolution we can use pola coodinates, so in vecto fom this is: t cos θ, t sin θ, t fo (t, θ), whee we have used t in place of to avoid confusion with the position vecto. Now detemine the nomal vecto: t cos θ, sin θ, 1, t sin θ, t cos θ,, N t t cos θ, t sin θ, t. The domain consists of the potion of the suface that lies below z 1, i.e. in the egion x +y 1. In pola coodinates (t, θ), this gives {(t, θ) R ; t 1, θ π}. Now to evaluate the suface integal: (x i + y j ) n d (t cos θ i + t sin θ j ) N dt dθ π 1 t dt dθ π 3.
Math 9 Assignment 9 olutions 4 7. Evaluate (x y i + xy j + z k ) n d whee is defined by z x y, z, and n is the unit nomal to the suface with negative z-component. olution We need a paametic epesentation of the suface. ince is a suface of evolution we can use pola coodinates, so in vecto fom this is: t cos θ, t sin θ, t fo (t, θ), whee we have used t in place of to avoid confusion with the position vecto. Now detemine the nomal vecto: t cos θ, sin θ, t, t sin θ, t cos θ,, N t t cos θ, t sin θ, t. The domain consists of the potion of the suface fo which z, i.e. in the egion x + y. In pola coodinates (t, θ), this gives {(t, θ) R ; t, θ π}. Now to evaluate the suface integal: (x y i + xy j + z k ) n d (t 3 cos θ sin θ i + t cos θ sin θ j + ( t ) k ) N dt dθ π π ( t 5 cos 3 θ sin θ t 4 cos θ sin θ t + t 3 ) dθ dt ( t 5 cos4 θ t 4 sin3 θ 3 (t 3 t) dt π. ) π + (t 3 t)θ dt 8. Find the centoid of the suface consisting of the pat of z x y above the xy-plane. olution We need a paametic epesentation of the suface. ince is a suface of evolution we can use pola coodinates, so in vecto fom this is: t cos θ, t sin θ, t fo (t, θ), whee we have used t in place of to avoid confusion with the position vecto. Now detemine the nomal vecto: t cos θ, sin θ, t, t sin θ, t cos θ,, N t t cos θ, t sin θ, t. The domain consists of the potion of the suface fo which z, i.e. in the egion x + y. In pola coodinates (t, θ), this gives {(t, θ) R ; t, θ π}. The aea of is given by: A() d N π dt dθ t 1 + 4t dt dθ 13π 3. By symmety, it follows that x ȳ. Thus, z-component of the centoid is: z 1 A() z d 1 ( t ) N dt dθ 1 A() A() Thus, the centoid is: ( x, ȳ, z) (,, 111 13 ). π t( t ) 1 + 4t dt dθ 7π A().
Math 9 Assignment 9 olutions 5 9. Find the moment of inetia about the z-axis of the suface consisting of the pat of z x y above the xy-plane. olution We need a paametic epesentation of the suface. ince is a suface of evolution we can use pola coodinates, so in vecto fom this is: t cos θ, t sin θ, t fo (t, θ), whee we have used t in place of to avoid confusion with the position vecto. Now detemine the nomal vecto: t cos θ, sin θ, t, t sin θ, t cos θ,, N t t cos θ, t sin θ, t. The domain consists of the potion of the suface fo which z, i.e. in the egion x + y. In pola coodinates (t, θ), this gives {(t, θ) R ; t, θ π}. The moment of inetia about the z-axis is: I z 1 (x + y ) d A() t N dt dθ π t 3 1 + 4t dt dθ 149π 13. 1. A cicula tube : x + z 1, y is a model fo a pat of an atey. Blood flows though the atey and the foce pe unit aea at any point on the ateial wall is given by F e y n + 1 j, y + 1 whee n is the unit oute nomal to the ateial wall. Blood diffuses though the wall in such a way that if d is a small aea on, the amount of diffusion though d in one second is F n d. Find the total amount of blood leaving the entie wall pe second. olution We need a paametic epesentation of the suface. ince is a cylinde with axis paallel to the y-axis, the most pudent way to paameteize the suface is by using pola coodinate θ in the xz-plane and y as the paametes, so in vecto fom this is: cos θ, y, sin θ fo (y, θ), whee {(y, θ) R ; y, θ π}. Now detemine the nomal vecto: y, 1,, sin θ,, cos θ, N t cos θ,, sin θ. The unit nomal n and suface element d ae given by: n N N cos θ,, sin θ, d N dy dθ dy dθ. The total amount of blood leaving the entie wall pe second is: F n d ( e y n + 1 ) j y n π dy dθ + 1 e y dy dθ π(1 e ).