Question AMA2112_20202_A/Midterm_March2021_v2.pdf 1/4 100% + Q1 Use spherical coordinates to evaluate the triple integral - y) dxdydz, where D is the solid region defined by the inequalities 72 + y2 + x2 <9, x > 0 and 2 2 0. [12 marks] Q2 Let C, be the line segment joining the origin and the point (1,1,2) and C, be the line segment joining the point (1,1,2) and the point (0,2,1). Integrate f(x, y, z) = (x + y)2 - 22 over CUC2 [13 marks] Q3 Let F = (y2 + y + cos x cos z)i + (2.xyz + e + sin 2)j + (xy + y cos 2 - sin x sin 2)k. (a) Calculate V.F and V X F. [5 marks) (b) Find f such that f= F. Hence, or otherwise, evaluate the line integral where T is given by r(t) = –2020 + cos(tºn)j + sin(-)k, Ost 51. [10 marks]
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Transcribed Image Text: AMA2112_20202_A/Midterm_March2021_v2.pdf 1/4 100% + Q1 Use spherical coordinates to evaluate the triple integral - y) dxdydz, where D is the solid region defined by the inequalities 72 + y2 + x2 <9, x > 0 and 2 2 0. [12 marks] Q2 Let C, be the line segment joining the origin and the point (1,1,2) and C, be the line segment joining the point (1,1,2) and the point (0,2,1). Integrate f(x, y, z) = (x + y)2 - 22 over CUC2 [13 marks] Q3 Let F = (y2 + y + cos x cos z)i + (2.xyz + e + sin 2)j + (xy + y cos 2 - sin x sin 2)k. (a) Calculate V.F and V X F. [5 marks) (b) Find f such that f= F. Hence, or otherwise, evaluate the line integral where T is given by r(t) = –2020 + cos(tºn)j + sin(-)k, Ost 51. [10 marks]
More Transcribed Image Text: AMA2112_20202_A/Midterm_March2021_v2.pdf 1/4 100% + Q1 Use spherical coordinates to evaluate the triple integral - y) dxdydz, where D is the solid region defined by the inequalities 72 + y2 + x2 <9, x > 0 and 2 2 0. [12 marks] Q2 Let C, be the line segment joining the origin and the point (1,1,2) and C, be the line segment joining the point (1,1,2) and the point (0,2,1). Integrate f(x, y, z) = (x + y)2 - 22 over CUC2 [13 marks] Q3 Let F = (y2 + y + cos x cos z)i + (2.xyz + e + sin 2)j + (xy + y cos 2 - sin x sin 2)k. (a) Calculate V.F and V X F. [5 marks) (b) Find f such that f= F. Hence, or otherwise, evaluate the line integral where T is given by r(t) = –2020 + cos(tºn)j + sin(-)k, Ost 51. [10 marks]
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Q1. ∭_(D)(x-y)dxdydz, where E is the solid inside the sphere x^(2)+y^(2)+z^(2)=9,x >= 0,z >= 0.In spherical polar coordinates (r,phi,theta) :By the relation between cartesian coordinates (x,y,z) and thespherical polar coordinates (r,phi,theta), we have x=r sin phi cos theta rarr(1);y=r sin phi sin theta rarr(2);z=r cos phi rarr(3), where r is distance of P(x,y,z) from O(0,0,0), phi is the angle made by vec(OP)_("with respect to positive "z"-axis and ") theta is the angle made by the projection of vec(OP) on the xy-plane with ... See the full answer