TUT 6 Mohu
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Institute of Chemical Technology Mumbai Tutorial - 6 MAT 1102: Applied Mathematics -II 1.(i) Find the volume of the prizm formed by planes aU : br,U : 0,r : &, z : 0, z : c + rA. (ii) Find the volume bounded by A2 : n,trz :3r and the planes z : 0 and, r * y * z : l. (iii) Find the volume of the prism whose base is the triangle in the ry-plane bounded by the x-axis, the line A:fr,r: l andwhose top lies in the plane z:3-r*A. (iv) A right circular cylindrical hole of radius b is board through the sphere of radius a. Find the volume of the remaining solid. (v) Find the volume bounded by the cone z2 : 12 *y2 and the paraboloid z: x)2 +y2. (vi) Find the volume bounded by cylinder 12 + y2: a2 and planes z :0, y + z : b. (vii) Find the volume of the cylind e, 12 + A2 : 2ar intercepted between the parabolo id. 12 + y2 : 2az and the rgr-plane. (viii) Find by double integration the volume of the sphere 12 +y2 + z2 : a2 att off by the plane z : 0 and the cylinder 12 * a2 : ar, 2.(i) The part of the lemniscate 12 :2a2cos2|,O < 0 < r14is revolved. about the r-axis. Find the surface area (s.a.) of the solid generated. (ii) The line segment r: sin2 t,y: cos2t,O < 0 < Tl2 is revolved about the y-axis. Find the surface area (s.a.) of the solid generated. (iii) Find the s.a, generated by revolving the curve (, -b)2 *y2: a2,b) o about y-axis. (iv) Find the s.a. generated by revolving the curve 12 1a2 +y2 lb2:l,a) b about r-axis. (v) The curve r: o,(1+cosd) revolves about the initial line. Find the s.a. (vi) A quadrant of a circle of radius o, revolves around its chord. Show that the s.a. of the spindle generated is ZJZra2(t - x). 3.(i) ABCD is a square plate of side o and O is the mid point of AB. If the surface density is proportional to the square of the distance from O, show that the center of gravity (C.G.) of the plate is at a distance 7all0 from AB. (ii) Show that the distance from the cusp of the C.G. of the cardioide r : a(1* cosd) when a density at a point varies as the square of the distance from the cusp is 6alb. (iii) Find the moment of inertia (M.I.) about the axes of the mass of a plate whose density p(r,a) is constant and is bounded by the curves A : 12 and y : r + 2. (iv) Find the M.L about tlie axes of the circular lamina 12 + A2 < o2 when the density p(r,y) is tFT?; (-o < r I a), (-JP=F < u s t/a41. 4. Problems on Fourier series, Fourier cosine series and Fourier sine series: See the examples (9.1 - 9.7) and evaluate all the problems (1-35) and (1-15), given in page: (9.13-9.15) and (9.21), respectively, (Book: Adv. in Eng. Mathematics, Jain & Iyengar,3rd edition). 5. Problems on Fourier integrals: See the example (9.10) and evaluate all the problems (1-15), given in page: (9.28), (Book: Adv. in Eng. Mathematics, Jain & Iyengar, 3rd edition). 6. Problems on Fourier transformq See the examples (9.21- 9.39) and evaluate all the problems (1-40) given in page: (9.71-9.73), (Book; Adv. in Eng. Mathematics, Jain & Iyengar, 3rd edition).
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Transcript of TUT 6 Mohu
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Institute of Chemical Technology MumbaiTutorial - 6 MAT 1102: Applied Mathematics -II
1.(i) Find the volume of the prizm formed by planes aU : br,U : 0,r : &, z : 0, z : c + rA.(ii) Find the volume bounded by A2 : n,trz :3r and the planes z : 0 and, r * y * z : l.(iii) Find the volume of the prism whose base is the triangle in the ry-plane bounded by thex-axis, the line A:fr,r: l andwhose top lies in the plane z:3-r*A.(iv) A right circular cylindrical hole of radius b is board through the sphere of radius a. Find thevolume of the remaining solid.
(v) Find the volume bounded by the cone z2 : 12 *y2 and the paraboloid z: x)2 +y2.(vi) Find the volume bounded by cylinder 12 + y2: a2 and planes z :0, y + z : b.(vii) Find the volume of the cylind e, 12 + A2 : 2ar intercepted between the parabolo id. 12 + y2 :2az and the rgr-plane.
(viii) Find by double integration the volume of the sphere 12 +y2 + z2 : a2 att off by the planez : 0 and the cylinder 12 * a2 : ar,
2.(i) The part of the lemniscate 12 :2a2cos2|,O < 0 < r14is revolved. about the r-axis. Findthe surface area (s.a.) of the solid generated.
(ii) The line segment r: sin2 t,y: cos2t,O < 0 < Tl2 is revolved about the y-axis. Find thesurface area (s.a.) of the solid generated.
(iii) Find the s.a, generated by revolving the curve (, -b)2 *y2: a2,b) o about y-axis.(iv) Find the s.a. generated by revolving the curve 12 1a2 +y2 lb2:l,a) b about r-axis.(v) The curve r: o,(1+cosd) revolves about the initial line. Find the s.a.
(vi) A quadrant of a circle of radius o, revolves around its chord. Show that the s.a. of the spindlegenerated is ZJZra2(t
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x).3.(i) ABCD is a square plate of side o and O is the mid point of AB. If the surface density is
proportional to the square of the distance from O, show that the center of gravity (C.G.) ofthe plate is at a distance 7all0 from AB.
(ii) Show that the distance from the cusp of the C.G. of the cardioide r : a(1* cosd) when adensity at a point varies as the square of the distance from the cusp is 6alb.
(iii) Find the moment of inertia (M.I.) about the axes of the mass of a plate whose density p(r,a)is constant and is bounded by the curves A : 12 and y : r + 2.(iv) Find the M.L about tlie axes of the circular lamina 12 + A2 < o2 when the density p(r,y) istFT?; (-o < r I a), (-JP=F < u s t/a41.
4. Problems on Fourier series, Fourier cosine series and Fourier sine series:See the examples (9.1
- 9.7) and evaluate all the problems (1-35) and (1-15), given in page:
(9.13-9.15) and (9.21), respectively, (Book: Adv. in Eng. Mathematics, Jain & Iyengar,3rdedition).
5. Problems on Fourier integrals:See the example (9.10) and evaluate all the problems (1-15), given in page: (9.28), (Book:Adv. in Eng. Mathematics, Jain & Iyengar, 3rd edition).
6. Problems on Fourier transformqSee the examples (9.21- 9.39) and evaluate all the problems (1-40) given in page: (9.71-9.73),(Book; Adv. in Eng. Mathematics, Jain & Iyengar, 3rd edition).
