SVKM s NMIMS. Mukesh Patel School of Technology Management & Engineering, Vile Parle, Mumbai

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2 Mukesh Patel School of Technolog Management & Engineering Page SVKM s NMIMS Mukesh Patel School of Technolog Management & Engineering, Vile Parle, Mumbai- 456 Tutorial Manual Academic Year : 4-5 Program: B. Tech / MBA (Tech) (All programs) Semester II Course: - Engineering Mathematics II

3 Mukesh Patel School of Technolog Management & Engineering Page SVKM S NMIMS Mukesh Patel School of Technolog Management & Engineering Course: Engineering Mathematics II

4 Mukesh Patel School of Technolog Management & Engineering Page Course Objectives: To provide an understanding of Engineering Mathematics with basic concepts and their application in technical subjects. Application of the concepts to solve engineering problems Course Outcomes: After the successful completion of this course, the student will be able to :. Solve problems using matri method.. Emplo Beta, Gamma techniques to solve integrals b selecting appropriate substitution.. Analse suitable method to solve differential equations. 4. Use concepts of multiple integrals to solve problems, compute area, mass of a lamina and volume.

5 Mukesh Patel School of Technolog Management & Engineering Page 4 LIST OF TUTOIALS S. NO. NAME OF THE TUTOIALS Page No. Tpes of Matrices, Adjoint and ank 6 Gamma Function and its application to solving Integrals 9 Beta Function and its application to solving Integrals 4 Eact Differential Equation, Non Eact Differential equation reducible to Eact form 5 Linear Differential Equations, Equations educible To Linear Form, Bernoulli s Equations 6 Linear homogeneous differential equation with constant coefficients, Nonhomogeneous linear differential equation with constant Coefficients 7 Method of Variation of Parameters, Equation reducible to linear differential equation with constant coefficients Evaluation of Double Integrals in Cartesian and Polar Coordinate sstems 4 9 Evaluation of Double Integration over a region 6 Double Integration b change of coordinate sstem and change of order 9 Applications of Double Integration Triple Integration 5

6 Mukesh Patel School of Technolog Management & Engineering Page 5 CO mapping with Tutorials Tutorial No. CO CO CO CO

7 Mukesh Patel School of Technolog Management & Engineering Page 6 Tutorial Tpes of Matrices, Adjoint and ank. Matrices AB, are such that A B= and 4A+ B= 4 Find AB, Given = + Find,, and z w w z+ w z w. Epress the following matrices as sum of smmetric and skew smmetric matrices ,, Epress the following matrices as sum of Hermitian and skew hermitian matrices. i i + i 4i i i i 4i 5, 5 i 8, i 4i 5 4+ i + i i i + i 6 4+ i + i i 5. Epress the following matrices as P+ iq where P and Q both are Hermitian matrices. + i i i i i i + i i i, + i + i, i + i + i + i + i i + i 5i + i i 6. Epress the following Hermitian matrices as P+ iq where P is real smmetric and Q is + i i i + i + i + i real skew smmetric. i i, i i, i i i i i + i i i 7. Epress the following Skew Hermitian matrices as P+ iq where P is real skew i i + i i + i i smmetric and Q is real smmetric. i i i, + i i + i + i i i i + i i i + i, i i i + i i i 8. Verif that adj(adj) A=A for A=. Does it happen for ever X matri? Eplain. 4 5

8 Mukesh Patel School of Technolog Management & Engineering Page 7 9. Find the adjoint and hence the inverse (if it eists): 8 4 4, 4 5. If A =, find A. Use adjoint method.. Find the condition on k such that the matri for k = 4 A= k 6 has an inverse. Obtain 5 A.. Find the matri A, if adj. A = 5. Verif that adj ( adj A) A A also find ( adj A) = where A =..,. ; 4. Prove that the following matrices are orthogonal and hence find A. cosα sinα A = sinα cosα ; A = 6 5. If a b A= c b 7 9 a c is orthogonal, find a, b, c. 6. educe the following matrices to ow Echelon form and find its rank. 4 4 a) 5 7 b) c) 4 d)

9 Mukesh Patel School of Technolog Management & Engineering Page 8 7. educe the following matrices to normal form and find their ranks: (i) (ii) 6 (iii) 4 4 (iv) (v)

10 Mukesh Patel School of Technolog Management & Engineering Page 9 Solve the following:. Find. Find Tutorial Gamma Function and its application to solving Integrals e e. Evaluate 4. Evaluate α 4 5. Prove that 6. Prove that 7. Prove that d d n h e d e d 8 4 e d e d = 6 e e d d = e 6 4 d e d 8. Evaluate ( 4) e d 9. Evaluate ( log ). Evaluate 4 d ( log ). Evaluate ( ). Evaluate. Evaluate 4. Evaluate 5 d 5 d log d log d log d =. 9

11 Mukesh Patel School of Technolog Management & Engineering Page 5. Evaluate dz 4 z 6. Prove that..5...(n ) = 7. Evaluate 8. Prove that cos a e a n d sin b d = n n + ab ( a + b )

12 Mukesh Patel School of Technolog Management & Engineering Page. Evaluate. Prove tat. Evaluate 4. Evaluate 5. Evaluate Tutorial Beta Function and its application to solving Integrals 9 (9 ) d d β, 5 = d d 8 d 6. Evaluate ( ) 7. Prove that 8. Evaluate 4 d 6 d ( ) d 4 4 d 9. Prove that ( ). Evaluate. Evaluate. Evaluate. Evaluate 4. Evaluate d = 4 a a d= a m d cos 4 sin d d d d 8

13 Mukesh Patel School of Technolog Management & Engineering Page 5. Evaluate ( + θ) 4 cos dθ 6. Evaluate ( + θ) 7. Evaluate 8. Evaluate cos4 dθ 4 7 cos d θ θ 6 cos sin 6 d θ 5 θ θ 9. Evaluate ( + ) 4 sin θ cosθ dθ. Evaluate ( + ). Evaluate. Prove that. Prove that 7 sin cos θ sinθ dθ cos 4 d p p+ sin d sin d =. 7 4 sin cos d 4. Evaluate 4 ( )( 7 ) 9 5. Evaluate 4 ( 9 )( 5) 5 d d 6. Evaluate 4 4 d Evaluate 9 6 d Prove that 9. Prove that ( ) 8 6 ( + ) 4 ( ) ( + ) d = 6 = 55 d = 5 55 ( p + ) d. Prove that =

14 Mukesh Patel School of Technolog Management & Engineering Page. Evaluate 4 5. Prove that + d ( + ) ( ) 7 d = β 4, ( + )

15 Mukesh Patel School of Technolog Management & Engineering Page 4 Tutorial 4 Eact Differential Equations, Non Eact Differential Equation reducible to Eact form Solve the following Differential equations :.. (sin cos ) (cos sin tan ) + e d+ + d = o ( 4 ) ( ) e + d+ e d= o sin = = 4. ( cos ) log ( 8) d d ; () d cos + sin = o d sin + cos + e 5. ( ) ( ) + d+ + d = o 6. cos d cos d d o + = 7. + e d+ e d = o given that () = cos d+ ( + log sin ) d = o log + + d d + = ( ). ( tan + sec ) = (tan ) d d Solve the following Differential equations : + d d = o. ( ) + + d+ + d = o 4. ( ) log d+ d = o. ( )

16 Mukesh Patel School of Technolog Management & Engineering Page 5 4. ( ) ( ) d d = o 5. ( ) d ( ) d = 6. (log ) d+ d = (4 + ) d + ( + ) d = ( e ) d d = ( e m ) d+ md=. 4 4 ( + ) d d = d+ + d =. ( ) ( ) Solve the following Differential equations : 4 4. ( ) ( 4 ) + d+ + d = o d + d = o. ( ) 4 4. ( ) ( ) e + + d + e d = o + e d e d = o 4. ( ) ( + ) d + ( + + ) d = 4 ( + d ) + ( ) d= e d ed ( + ) = ( + e) d ( e+ ) d= 9. e ( d d) + e d + e d = d + = d. ( ) Solve the following Differential equations :. ( ) ( ) + + d + d = o

17 Mukesh Patel School of Technolog Management & Engineering Page 6. ( sin cos ) ( cos sin ) + d+ d = o d d ( + ) + ( ) = d d ( + ) + ( + + ) = d + = d d d ( + + ) + ( + ) = 7. ( + ) d ( ) d = 8. d d ( + ) + ( + ) = 9. d d ( ) + ( + ) =. ( ) ( ) + d d= o Solve the following Differential equations :. ( ) ( ) d d= o. ( ) ( ) + d d= o d d = o. ( ) + + d d = o 4. ( ) ( + ) d ( + ) d = ( + ) d d = d d ( + ) = ( ) d ( ) d = ( + ) d d = d d ( ) + =

18 Mukesh Patel School of Technolog Management & Engineering Page 7 Tutorial 5 Linear Differential Equations, Equations educible To Linear Form, Bernoulli s Equations Solve the following Differential equations :. d ( ) d = o. ( + ) = ( tan ) d d. dr + (r cotθ + sin θ) dθ = d 4. sin tan d = + d 4 +. = d + ( + ) 5. d 6. cos ( sin cos ) d + + = d 7. + = d d ( ) + = d d d ( ) ( ) = ( ) ( + + ) d+ ( + ) d= tan. ( + ) = ( ) d e d. ( + ) d+ ( + ) e d= d d. (+ sin ) = [ cos (sec + tan ) ] Solve the following Differential equations :. d d + = sin cos

19 Mukesh Patel School of Technolog Management & Engineering Page 8. d tan tan cos.cos d + =. e ( + ) d+ ( e e ) d = 4. d = e.( e e ) d 5. d sin sin d + + = 6. d + d + = 7. d 4 + = d 8. d ( ) d = + 9. d tan = ( + e ) sec d + d. sec sin cos d + = (tan )( ) d tan tan cos d + =. ( ) d e. = d Solve the following Differential equations :. d 6 d + = d + = d d 4 d = ( ). ( ) ( ) e d d = d d = cos = ( sin )cos d d

20 Mukesh Patel School of Technolog Management & Engineering Page 9 d ( 7. = + e /).log d d 8. + = d d 9. + = d. d + ( ) d = d. sin d = d. 4 ( ) ( ) d = + + +

21 Mukesh Patel School of Technolog Management & Engineering Page Tutorial 6 Linear homogeneous differential equation with constant coefficients, Non-homogeneous linear differential equation with constant Coefficients Solve the following differential equations : d d + 5 = d d d d d dt dt dt = d d d d + = 4 d d + + = 4 d d d 8 d + = d 4 d d + = 4 d d d ( D ) ( D + D+ ) = 4 7. { } Solve the following differential equations :.. d d + + = e d d d d = e + d d D D 5D+ 6 = e + 8. ( ) 4. ( + ) = 5. ( + + ) = 6. ( ) sin D D cos 4 D D 6 sin D D + = + e 7. ( D 4 + D + 9) = cos ( + ) 8. ( D + D) = cos

22 Mukesh Patel School of Technolog Management & Engineering Page 9. 4 d 4 d + = cos ec cos ec. sin. ( ). ( ) D + D + D = + D + 4 = + sin. ( D 4D+ 4) = 8( + sin+ e ). ( D + 5D+ 6) = + e 4. d d + 4 = 5+ d d 5. ( D D 6D) = + 6. ( D 4D+ 4 ) = e. 7. ( 7 6) = ( + ) 8. ( D D ) = e + 9. ( ). D D e 7 6 ( ) D D + D = e + e d cos e e d + = +. ( ) =. ( ) =. ( ) = 4. ( ) 5. ( + ) = 6. ( 4 ) = 7. ( ) D coshcos D sin D sin 4 D + D + = sin 4 D 4 sin D 4 sinh D + D+ = + e D D = e sin 8. ( ) D = e sin 9. ( ). d d + + = e + 4 d d

23 Mukesh Patel School of Technolog Management & Engineering Page Tutorial 7 Method of Variation of Parameters, Equation reducible to Linear Differential Equation with constant coefficients Solve following differential equations using variation of parameters method :. d k tan d + =. ( D ). ( ) = + e k D = e sin e D D+ = 4. ( 6 9) 5. d a sec d + = a Solve following differential equations : d d + 5 = sin(log ) d d d d + = d d d d d + = 8+ 4log (Hint: multipl throughout b to get Cauch s D.E.) d d d + + = d d + d d d + + = log (Hint: multipl throughout b to get Cauch s D.E.) d d d d d + + = d d ( + )

24 Mukesh Patel School of Technolog Management & Engineering Page Solve following differential equations : (5 + ) d d 6(5 ) 8 d + d + = d d ( + a) 4( + a) + 6 = d d d d (+ ) (+ ) = 9 d d d d (+ ) (+ ) = 6 d d ( ) d d + ( ) 4 d + d + = +

25 Mukesh Patel School of Technolog Management & Engineering Page 4 Tutorial 8 Evaluation of Double Integrals in Cartesian and Polar Coordinate sstems. Evaluate the following Double integrals (Cartesian Coordinates): i) ii) iii) e dd ( ) a a + dd dd iv) cos( + ) v) 5+ dd vi) ( + ) vii) viii) i) ) i) ii) iii) dd dd cos sin a dd ( + ) ( + ) a dd e dd ( a )( ). dd d d a + a dd + + a ( + ) dd iv) v) vi) a e dd ( a )( ) dd 4 4 a 5a a/ a + d d vii) log ( + ) viii) i) ) a + + d d ( ) + a a+ a dd ( 4a + + ) 4a ( ) d d ( + ) 4a d d

26 Mukesh Patel School of Technolog Management & Engineering Page 5. Evaluate the following Double integrals (Polar Coordinates): i). ii) iii) iv) v) acosθ r a r drd a ( + sinθ ) 4 cos θ r acosθ θ. cosθdrdθ r ( + r ) r 4sinθ rdrd sinθ drdθ sinθdrdθ θ

27 Mukesh Patel School of Technolog Management & Engineering Page 6 Tutorial 9 Evaluation of Double Integration over a region. Evaluate dd over the region given b. Evaluate ( + ) + = a, and. dd throughout the area enclosed b the curves = 4, + =, = and =. + dd over the region area of the triangle whose vertices are. Evaluate ( ) (, ),(, ),(, ). 4. Evaluate dd over the region given b 5. and = 4 = 4. d d ( ) ( ) where is the region in the first quadrant bounded b, and =, + =. 6. ( ) d d where is the region bounded b, = 4, =, = and = 4. d d over + 7., 8. dd is region in first quadrant bounded b 6 =, = 8, = and = 9. ( + ) dd is region bounded between = and =... dd where is the region bounded b dd where is the region bounded b dd =, =, =. =, + =.. where is the +ve quadrant of the circle + =.. Prove That: dd = 6 where is triangle having vertices (,), (,) & (,).

28 Mukesh Patel School of Technolog Management & Engineering Page 7 4. dd where is bounded b = & = dd where is bounded b = a = a 4 & 4. is region bounded b 4,,, 4 6. ( ) d d = = = =. 7. Evaluate dd over the region which is the first quadrant of the ellipse 8. + =. dd where is bounded b + = = =, &. 9. e dd where is triangle formed b (,),(,),(,). ( )( ). sin ( a + b) dd where is bounded b =, =, a + b =.. Evaluate r drdθ where A is a loop of of A r + 4. Evaluate rsin θ drdθ. Evaluate 4. Evaluate 5. Evaluate r = 4cosθ. over the area of the cardiode r a( cosθ ) = +. r a r drdθ over the upper half of the circle cos rdrdθ r r = a θ. where is area common to the circles = sinθ, r = cosθ. rdrdθ where is area between r = + cosθ, r =. 6. Evaluate sinθda where is the region in the first quadrant that is outside the circle r = and inside the cardioid r = ( + cosθ ). rdrdθ 7. Evaluate where is area bounded r 8. Evaluate r = a cos θ. ( + r ) ( a + r ) = cos θ. rdrdθ where is area bounded within one loop of the lemniscate

29 Mukesh Patel School of Technolog Management & Engineering Page 8 9. Evaluate. Evaluate rdrdθ over the area included between the circles r = acos θ, r = acosθ. 4 cos r θ drdθ where is the interior of the circle r = acosθ.

30 Mukesh Patel School of Technolog Management & Engineering Page 9 Tutorial Double Integration b change of coordinate sstem and change of order. Change to polar coordinates and evaluate the following integrals: i) ( + ) a dd ii) log ( + ) a a dd iii) + iv) v) vi) vii) viii) a a a e dd 4 + dd dd dd e ( ) e dd dd i) ) i) ii) + dd ( ) + 4a 4a ( + ) dd ( + ) a dd ( + ) a a dd iii) sin ( ) iv). Transform the integral to Cartesian form and hence evaluate ( ) a a e a a dd dd a r sinθcosθdrdθ. +. Transform to polar and evaluate dd where is bounded b + = b, a,b >. 4. Transform to polar and evaluate ( ) circle + = a. + = a, + dd where is the first quadrant of the

31 Mukesh Patel School of Technolog Management & Engineering Page 5. Transform to polar and evaluate ( ) 6. Transform to polar and evaluate sin ( + ) + dd where is bounded b dd where is bounded b + = a. + = a. 7. Transform to polar and evaluate + dd a b where is in first quadrant bounded b + =. a b m n 8. Transform to polar and evaluate ( ) dd where is in first quadrant bounded b a + =. b ab b a 9. Transform to polar and evaluate dd over ellipse ab + b + a. Evaluate dd over circle =.. Evaluate dd + where is bounded b ( ) + = a, a + =. b + = b, (a>b).. Transform to polar and evaluate dd over one loop of lemniscate + + ( ) + =. ( ) dd. Transform to polar and evaluate where is bounded b 4. Change the order of integration: + =,. 4 i) f (, ) 4 a / ii) f (, ) a a dd dd iii) f (, ) dd bc / iv) f (, ) a v) f (, ) a dd dd vi) f (, ) dd

32 Mukesh Patel School of Technolog Management & Engineering Page vii) f (, ) 6 5+ viii) f (, ) k k+ k dd dd i) f (, ) k k dd 4 9 ) f (, ) a + a dd i) f (, ) a ii) f (, ) dd dd 5. Change the order of integration and evaluate the following : i) ii) a d d ( + a) ( a )( ) 4 cos iii) ( + ) iv) v) vi) vii) a a a a a dd dd d d b a b a a a b b ( ) d d + e a d d d d viii) i) ) i) ii) iii) iv) e dd d d d d 4 4 d d ( + ) ( + ) a d d d d ( a )( a )( ) cos sin a dd 6. Epress as a single integral and then evaluate ( + ) dd+ ( + ) dd.

33 Mukesh Patel School of Technolog Management & Engineering Page Tutorial Applications of Double Integration. Find the area bounded b the lines = +, = +, = 5.. Find the area bounded b ellipse. Find the area between parabola a + =. b =, the line = & -ais. 4. Find the area outside the parabola = 4 4 & inside = Find the area between the curves = a and =, a>. a 6. Find the area of the curvilinear triangle ling in first quadrant, bounded b = 4a& inside + = a. =. 7. Find the area of the loop of the curve ( )( ) 8. Find the area bounded between the curve ( ) + a = a and its asmptote. = 4a, 9. Find the area between the rectangular hperbola = and the line + = 6.. Find the area bounded b the parabola = 4aand its latus rectum. =.. Find the area of the loop of the curve ( )( ). Find the area bounded between the curve ( ) + a = a and its asmptote.. Find the area between the rectangular hperbola = and the line + = Find the area bounded b the curves = & line + =. 5. Find the total area of the astroid. + = a 6. Find the area enclosed b one loop of + =, b converting it to polar 4 4 a coordinates. 7. Find the area common to r = acosθ & r = a. 8. Find the area common to r = acosθ & r= asinθ.

34 Mukesh Patel School of Technolog Management & Engineering Page 9. Find the total area included between the two cardioids r = a( + cos θ) & r = a( cos θ).. Find the area ling inside a cardioid r = + cosθ and outside r( + cos θ ) =.. Find b double integration the area inside the circle r = asinθ & outside the cardioid r = a( cos θ ).. Find area of the curve r = acosθ.. The larger of the two areas into which the circle r = 6 is divided b the parabola = In the ccloid = a( θ + sin θ), = a( cos θ).find the area between the base and the portion of the curve from cusp to cusp. 5. A lamina is bounded b the curves = & =.If the densit at an point is given b λ. Find the mass of the lamina. 6. Find the mass of the lamina in the form of an ellipse a + =. If the densit at an b point varies as the product of the distance from the aes of the ellipse. 7. The densit at an point of a cardioide r = a(+ cos θ ) varies as the square of its distance from its aes of smmetr. Find its mass. 8. Find the mass of the plane in the form of one loop of lemniscate r = a sin θ, if the densit varies as the square of the distance from the pole. 9. The densit at an point of a uniform circular lamina of radius a varies as its distance from a fied point on the circumference of the circle. Find the mass of the lamina.

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36 Mukesh Patel School of Technolog Management & Engineering Page 5 Tutorial Triple Integration Evaluation in Cartesian Coordinates. Evaluate log + e + + z. Evaluate dddz. z+ z dddz z dzdd.. Evaluate ( + + ) z. dddzthroughout the volume of the tetrahedron 4. Evaluate ( + + z) dddz where :,, z 5. Evaluate z,, z and + +. a b c Evaluation in Spherical Coordinates 6. Evaluate 7. Evaluate 8. Evaluate ( a r ) / asinθ a rdθdrdz dzdd b changing to spherical polar coordinates. z dddz ( ) V + + z + + z = 6 and 9. Evaluate ( ). Evaluate. Evaluate where V is the region bounded b the spheres + + z = 5. m + + z dddz, m > over the unit sphere. coordinates. zdddz dddz over the hemisphere over the volume of the ellipsoid z z a, + +. z + + = using spherical a b c

37 Mukesh Patel School of Technolog Management & Engineering Page 6. Show that if is the ellipsoid z a b c + + = abc abc bc ca abzdddz=. 4 Evaluation in Clindrical Coordinates. Evaluate ( + ) V 4. Evaluate ( + + ) Volume plane z = and z= h. dddzwhere V is the volume bounded b z z dddzover the volume of the + = and z =. z + = a intercepted b the 5. Find the volume of the tetrahedron bounded b the coordinate planes and the plane z + + =. a b c 6. Find the volume bounded b the clinders z =, + + z =. 7. Find the volume enclosed b the sphere 8. Find the volume of the paraboloid =, = and the planes + + z = 6 and the cone + = cut off b the plane z = 4. 4z 9. Find the volume of the solid bounded b the plane z =, the paraboloid and the clinder + = 4.. Find the volume of the ellipsoid : z a b c + + =. + = z tanα. + + = z

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