UNIVERSITI TEKNOLOGI MALAYSIA SSCE 1993 ENGINEERING MATHEMATICS II TUTORIAL 2. 1 x cos dy dx x y dy dx. y cosxdy dx

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1 UNIVESITI TEKNOLOI MALAYSIA SSCE 99 ENINEEIN MATHEMATICS II TUTOIAL. Evaluate the following iterated integrals. (e) (g) (i) x x x sinx x e x y dy dx x dy dx y y cosxdy dx xy x + dxdy (f) (h) (y + x)dy dx (j) x / sinx cosy x ( y ) x cos dy dx x y dy dx x sin y dxdy e y/x dy dx x sin y dxdy. Evaluate the following integrals. (x y )da, : The region bounded by y = x, y = + x and y =. (e) (f) y da, cos(y )da, : The region in the first quadrant of xy-plane, bounded by the curve y = x, the line y = x and the x-axis. : The region bounded by y = x, the line y = and y-axis xda, : The region bounded by the curve y = x, the lines y = x and x =, and x-axis. x da, : The region bounded by y = 6 and the lines x = 8 and x y = x. e x da, : The region bounded by y = x, the line x =, the x-axis and the y-axis.. Find the areas of the following regions using double integral. The region bounded by the curves y = sinx, y = cos x between x = and x = /. The region bounded by the curve y = x, the line y = x, y = and y =. The region bounded by the curves y = 9x and y = x 9.

2 SSCE 99: Tutorial. Evaluate the following integrals by first changing the order of integration. (e) y cos(x )dxdy siny dy dx x y / / sin(y )dy dx x (f) y 8 x x e xy dxdy x y + x 5. Evaluate the following integrals using polar coordinates. (e) x (x + y )dy dx y cos(x + y )dxdy y ( + x y + y )dxdy (f) x x dy dx ( y)e x+y dy dx y dy dx y y( y) xy dy dx x + y y x + y dxdy 6. Find the area of the region in the first quadrant bounded by the circle r = sin θ. 7. Evaluate sin θ da, : The region in the first quadrant bounded by the circle r = and the cardioid r = ( + cos θ). 8. Find the volume of the solid bounded by the surface r + z =. 9. Find the area of the region bounded by r = sinθ.. Evaluate e (x +y ) da, : The region bounded by the curve y = x and the x-axis.. Evaluate y da, : The region in the first quadrant bounded by the line y = x and the circle (x ) + y =.. Evaluate da, : The region in the first quadrant bounded by the + x + y lines y =, y = x and the circle x + y =.. Use double integrals to find the volumes of the following solids. The solid in the first octant bounded by x + 6y + z = and the coordinate planes. The solid bounded by the planes z =, x =, x =, y =, y = and the surface z = x + y. The solid bounded by the cylinder x + y = 9, and the planes z = and x + z = 5.

3 SSCE 99: Tutorial. Find the centroid of the lamina with uniform density δ(x, y) = bounded by the parabola y = x and the line x + y =. 5. Find the center of gravity of the lamina with density δ(x, y) = x + y, satisfying the inequalities x + y 9 and y. 6. A lamina on the xy-plane has the shape of a semi-circle x + y, y. Find the centre of mass if the density at each point is proportional to its distance from the origin. 7. Find the moment of inertia I x of the lamina which is bounded by the graph y = x and x-axis and has density δ(x, y) = x. 8. Evaluate the following iterated integrals. cosz yz cos(xy)dz dxdy 6 y y dxdy dz x lnz x xe y dy dz dx 9. Evaluate the following integrals. z dv, : The tetrahedron in the first octant bounded by x+y+ z =. xy sin(yz)dv, : The rectangular box bounded by x, y and z /6. xyz dv, : The solid in the first octant bounded by the parabolic cylinder z = x, planes y = x, z = and y =.. Find the volumes of the following solids using triple integrals. The solid bounded on its sides by the surface y = x, above by the plane y + z = and below by the plane z =. The solid in the first octant bounded by the coordinate planes, the plane y+z = and the parabolic cylinder x = y. The solid bounded by the parabolic cylinder z = y and rectangular planes x =, x =, y = and y =.. Use cylindrical coordinates to evaluate the following integrals. z x + y dv, : the solid bounded by surfaces z = and z = x + y. z dv, dv, : the cylinder y + z = which intersects with the planes y = x, x = and z =. : the solid bounded by paraboloids z = 8 x y and z = x + y.

4 SSCE 99: Tutorial. Find the volume of each of the following solid using cylindrical coordinates. The solid bounded by the paraboloid z = r and the plane z = 9. The solid bounded below by the plane z =, on its side by the cylinder r = sinθ and above by z = r. The solid bounded by the cylinder x + y = 9, and the planes x + z = 5 and z =.. Evaluate the following integrals using cylindrical coordinates. y y x y (x + y )dz dxdy, 9 (x +y ) y dz dxdy x 6 (x +y ) x dz dy dx. Find the volume of the following solid using spherical coordinates. The sphere x + y + z = 9. The solid bounded above by the sphere ρ =, and below by the cone φ = /. The solid that lies above the xy-plane, outside the cone z = x + y and inside the sphere x + y + z = Evaluate the following integrals using spherical coordinates. x x x x x y dz dy dx x y x y z x + y + z dz dy dx dv x + y + z, : sphere x + y + z 9 6. Find the centroid of the solid that lies below the sphere ρ = and above the cone φ = /. 7. Find the centroid of the solid bounded by the surface z = r and the plane z = Find the moment of inertia of the solid created by intersection of the sphere ρ with the cone φ = / and z-axis if the density of the solid is δ(x, y, z) =. 9. Session / Sem II A lamina in the first quadrant is bounded by x-axis, y-axis and x + y = 9. If the density of the lamina is δ(x, y) = e x +y, use polar coordinates to find the mass of the lamina.

5 SSCE 99: Tutorial 5 Use cylindrical coordinates to evaluate y Use spherical coordinates to evaluate x + y + z dv. Session /5 Sem I (x +y ) y dz dxdy. Find the volume of the solid bounded by z = y, y = x, the xy-plane and the yz-plane. iven that is a solid in the first octant bounded by x + y =, y + z = and z = x + y. If the density is δ(x, y) = x, use cylindrical coordinates to find the mass of the solid. Use spherical coordinates to evaluate x + y + (x + y + z ) dv where is the solid bounded above by the sphere x + y + z = and below by the cone z = (x + y ).. Session / Sem I Find the center of mass of a lamina bounded by y = x and y = with density δ(x; y) = y. Evaluate the area in the first quadrant bounded by the curves r =, r = sinθ, and r = cos θ. (7 mark) By using the spherical coordinates, evaluate z dv where is the solid bounded below by a cone z = x + y and above by a sphere x + y + z = z.. Session / Sem II Find the moment of inertia about the x-axis, I x, of a lamina bounded by a parabola x = y, the line y = x and the x-axis in the first quadrant with density δ(x; y) =.

6 6 SSCE 99: Tutorial By using the cylindrical coordinates, evaluate x + y dv where is the solid bounded below by the plane z = and above by the plane z =, and it s sides by the surface x + (y ) =. By using the spherical coordinates, evaluate. Session 5/6 Sem II 5 5 x 5 x +y dz dy dx. (7 Marks) Evaluate (x + y)da, given that is the region inside a triangle with vertices (, ), (, ), and (, ). ( marks) A circular sector lamina of radius r with an angle β, is as shown in Figure. y.... r β Figure..x If the lamina has constant density, use double integral in polar coordinates to answer the followings: (i) Show that the area of the lamina is r β. (β in radians). (ii) Find the centroid of the lamina. (8 marks) Transform the integral x x 6 x y dz dy dx into each of the following coordinates systems: (i) cylindrical, (ii) spherical. Hence, evaluate the integral using only one of the coordinates system. (8 marks). Session 6/7 Sem I iven that is a region on the xy-plane bounded by the lines y = x, y =, y = and the y-axis. (i) Sketch the region.

7 SSCE 99: Tutorial 7 (ii) Use polar coordinates to evaluate x + y da. Evaluate the integral z dv, where is the tetrahedron in the first octant bounded by the plane x+y+z = 6. Find the volume of the solid bounded above by the cone z = x + y, below by the xy-plane and laterally by the cylinder x + y = y. 5. Session 6/7 Sem II iven the following integral x f(x, y)dy dx. Sketch the region of integration. Obtain the equivalent integral in the reversed order. (5 marks) Use cylindrical coordinates to find the volume of the solid below the surface x + y + z = and above the plane z =. Use spherical coordinates to evaluate y x y y 6. Session 8/9 Sem I x +y ( x + y + z ) dz dxdy. (8 marks) Using double integrals in polar coordinates, find the mass of a semi-circular lamina x + y =, with y, and density function given by δ(x, y) = x y. Evaluate the following triple integrals in an appropriate coordinate system: (x + y ) dv, where is the solid in the first octant bounded by paraboloids z = x +y and z = x y. Evaluate the following iterated integrals by changing the coordinate system: 6 x x +y x + y + z dz dy dx.

8 8 SSCE 99: Tutorial 7. Session 8/9 Sem II Evaluate the integral ( x + y ) da, where is the region in the first quadrant bounded above by the circle x +y = 8 and below by the line y = x. Find the volume of the solid below the cone z = x + y, above the xy-plane and inside the cylinder x + y =. Find the mass of the solid in the first octant bounded by x + y + z = and the coordinate planes with the density function δ(x, y, z) =.. e ANSWES TO TUTOIAL 8 5 (e) 6 sin (f) (g) ln (h) (e ) (i) 5 (j) ln (e) 576 (f) (e ) sin (e ) ln7 (e) (f) e 5. 8 sin (e) (f) ln5.. 6 ( e ). 6

9 SSCE 99: Tutorial 9 (., ) (, ) 5 6. (, ) ( ) ( 6.,, 9 ) (e9 ) 6 ( 9,, x =, ȳ = (i) (ii) 6 r / secφ ) (ii) x = r β sinβ, r dz dr dθ ρ sin φdρdφdθ = ( ) 6 ln7 5 ȳ = r [ cos β] β. (ii) ln y y f(x, y)dxdy [ ] [ ] = 6.7