UNIVERSITI TEKNOLOGI MALAYSIASSE 1893 ENGINEERING MATHEMATICS TUTORIAL 4 1. Evaluate the following line integral Z x2 y ds, C : curve x = 3 cos t, y = 3 sin t, 0 ≤ t ≤ 21 π. (i) ZC (ii) (sin x + cos x) ds, C : line segment from (0, 0) to (π, 2π). ZC (iii) (2x + 9z) ds, C : curve x = t, y = t2 , z = t3 , 0 ≤ t ≤ 1. C Z (y − x) dx + x2 dy, C : curve y = x2 from point (0, 0) to (4, 16). (iv) C Z (v) (x2 + y 2 ) dx + 2xy dy, C : curve x = t2 , y = t3 , 0 ≤ t ≤ 32 . C Z (vi) (x + y + z) dx + x dy − yz dz, C 2. C : line segment from (1, 2, 1) to (2, 1, 0). Evaluate the line integral Z C F . dr for the given function F and curve C. (i) F(x, y) = x3 y i + (x − y) j, (ii) F(x, y) = (x2 − y 2 ) i + x j, (iii) F(x, y) = (x3 − y 3 ) i + xy 2 j, C : parabola y = x2 from point (−2, 4) to (1, 1). C : curve r(t) = t2/3 i + t j, −1 ≤ t ≤ 1. C : curve x = t2 , y = t3 , −1 ≤ t ≤ 0. (iv) F(x, y, z) = 2x i + xz j + yz k, C : line segment from (2, 2, 1) to (0, 2, 1) to (0, 0, 3). (v) F(x, y, z) = yz i − xz j + xy k, C : curve x = et , y = e3t , z = e−t , 0 ≤ t ≤ 1. (vi) F(x, y, z) = z i + x j + y k, C : curve x = sin t, y = 3 sin t, z = sin2 t, 0 ≤ t ≤ 21 π. 3. Show that the line integral is independent of path and evaluate the line integral by choosing a convenient path or by using a potential function. Z y 2 dx + 2xy dy, C : line segment from (1, 2) to (i) C (0, 0) to(1, 3). Z (y 2 + 2xy) dx + (x2 + 2xy) dy, (ii) C Z C : line segment from (−1, 2) ke (0, 3) to (3, 1). (6xy 3 + 2z 2 ) dx + 9x2 y 2 dy + (4xz + 1) dz, C : line segment from (0, 0, 0) ke (1, 0, 0) to (1, 1, 0) to (1, 1, 1). Z (iv) (y + z) dx + (x + z) dy + (x + y) dz, C : line segment from (0, 0, 0) ke C (0, 1, 0) to (−1, 0, π). (iii) C (i) Show that for I curve C with the parametric equations x = cos t. F(x. 0. C : line segment from (0. 0). 0 ≤ t ≤ 2π. dr is independent of path. 0) to (1. 0) to (1. y) = ey i + xey j. 6. 2. y = b sin t. 2 + y2 x x + y2 C Why is this result different from (i)? 7. F(x. 0) to (1. y) = (−3y + 2xy 2 ) i + 3xy j C : curve x = 4 cos t. y) = y 3 i + x3 j IC C : curve r(t) = 2 cos t i + 2 sin t j. C : line segment from (0. 1. 0) to (i) C (0. 0. x = 2 dan y = 14 x3 . 0 ≤ t ≤ 2π. y) = 2 .2 SSM2083/2283: Tutorial 4 4. 2) to (2. 1. y = sin t. 1). (ii) (iii) (iv) (v) (vi) 5. 0). C : curve y = 1 − x2 from point (0. z) = 8xz i + (1 − 6yz 3 ) j + (4x2 − 9y 2 z 2 ) k. C 0 ≤ t ≤ 2π. 1) to (0. C : curve r(t) = (t + sin t) i − t cos t j. y = sin t. 0 ≤ t ≤ 2π. C : line segment from (1. F(x. F . y) = (10x − 7y) i − (7x − 2y) j. Evaluate the following integral using Green’s theorem. 0) to (1. dr. I (ii) C (iii) (iv) (v) (vi) I I I 3y dx − x2 dy. F(x. 0) to (1. 2 C By using this result show that the area of an ellipse with the parametric equations x = a cos t. 0. Show that Z F . 2). C : circle x = cos t. C : line segment from (0. y. 0) to (1. dr. then 2 + y2 x x + y2 C ∂g −y x ∂f = for f (x. F(x. C : line segment from (0. 1) to (0. 3) to (0. 1) to (0. (2x + y 2 ) dx + (x2 + 2y) dy. 0 ≤ t ≤ π. C : boundary of the region bounded by y = 0. F . −y x dx + 2 dy = 2π. 0 ≤ t ≤ 2π is πab. 0) to (1. 0) to (−1. 0. (i) F(x. C C F . y. 0) to (1. z) = 3x2 i + 6y 2 j + 9z 2 k. 1. y) = (ex − y 3 ) i + (cos y + x3 ) j. The area of region R bounded by the curve C is given by the line integral ¸ ·I 1 −y dx + x dy . 2. dr. y. y) = y sin x i − cos x j. F(x. I x2 y dx + x dy. z) = yz i + xz j + xy k. 1) to (1. . F(x. 1). √ C : hemisphere y = 1 − x2 from (1. 0. y = 4 sin t. (ii) Show that 2 ∂y ∂x x + y2 Ix + y −y x Using Green’s theorem show that dx + 2 dy = 0. 0). 0). F(x. y) = 2 and g(x. Hence find a potential function and evaluate the line C integral along C. 1) (6 Marks) b) Show that the vector function F(x. 0. Compute the work done if the motion starts from point P (2. Given the force field. −2. 0. 0) to point Q(−2. 1. y. y. 1). By using this value of λ. 0) along the hemisphere with the parametric equations x = 2 cos θ. Show that the line integral C F . z) = (x +y ) i + (λxy) j + z k. 0) to the point (1. 9. 0). 1) to B(2. 0) to (1. Session 2001/02 Sem I a) Evaluate the line integral Z C (x2 + y 2 − x)dx + y p x2 + y 2 dy with C is the path consist of a line √ segment from (0. y. (1. 0) along a semicircle y = 1 − x2 . 1. is independent of the path if λ = 2. dr with F(x. y. 0) to (−1. (7 Marks) c) By using the Green’s theorem. 2 2 R (10 Marks) 12. y. 0). from (0. 1. evaluate I (xy + y 2 )dx + (x2 )dy C if C is the boundary of a region bounded by y = x2 and y = x in the anticlockwise manner. z) = (3x2 +6y) i − (14yz) j + (20xz 2 ) k. 0) and from (2. Session 2002/03 Sem I R a) Evaluate C F . 0) along the semicircle y = 4 − x2 on the xy−plane. 0) to (1. 1. 11. . 0. z) = (x2 + y) i − (3xz) j + (z − 4y) k. (10 Marks) b) Given F(x. z) = x2 z i − yx2 j + 3xz k acting on an object moving along the straight line from point (1. 0) to (−2.SSM2083/2283: Tutorial 4 3 8. y = 2 sin θ. 1. and from (1. 1. 4). and from (1. 0) to (1. 0) to (1. 0. evaluate the above line integral if C is the path that connects the points A(1. 0). with the path C as the line segment√that connects the point (1. y) = (x2 − y) i + x j acting on an insect as it moves along a circle with radius 2. Compute the work done by the force field F(x. evaluate the line integral C F . y) = (ey + yex ) i + (xey + ex ) j. 0 ≤ θ ≤ π. 0) to (1. 0. (7 Marks) 13. 2) to (2. 0. Session 2001/02 Sem II R a) If F(x. 0. 1. 0. Compute the work done by the force F(x. Hence find the work done in this field to move an object from a point (1. 10. F(x. dr. 1) with the path C as the line segment that connects the point (0. 1. 0) along the x−axis. 3. 1) to (0. 0. dr. 1) to a point (3. z) = (2xy + z 3 ) i + (x2 ) j + (3xz 2 ) k is a conservative field and find the potential function for F. y. show that the area of a circle with radius a is πa 2 . 3. 1. 4). Session 2002/03 Sem II R a) Evaluate C F . Hence find the work done in this field to move an object from a point (1. z) = (ye−x ) i + (zey − e−x ) j + (ey ) k is a conservative field and find the potential function for F. evaluate I (2xy 3 )dx + (4x2 y 2 )dy C if C is the boundary of a region bounded by y = x + 2. then 1 2 I C x dy − y dx = Area A Hence by using this formula. dr with F(x. If it is a conservative field. 0. x−axis and x = manner.4 SSM2083/2283: Tutorial 4 (7 Marks) b) Show that the vector function F(x. 1) to (1. z) = ey+2z [i + x j + 2x k] is a conservative field. 1) to B(7. (5 Marks) b) Determine whether the vector function F(x. 0) to (0. with the path C as the line segment that connects the point A(1. 3) to (3. y. obtain the potential function for F. 4). (7 Marks) c) By using the Green’s theorem in the xy−plane. 1. √ y in the anticlockwise (6 Marks) 14. show that if A is the area of the region bounded by a simple closed curved C. 3. y. Session 2005/06 Sem I R a) Evaluate C xy 2 dx + xy 2 dy along the path C = C1 ∪ C2 as shown in the Figure A. 2) to (3. (8 Marks) 15. 2. (7 Marks) c) By using the Green’s theorem. 0. 1). −1) to (0. 1. Hence find the work done in this field to move an object from a point (1. 5). Also evaluate this integral along the straight line path C3 from (0. z) = (y 2 + z) i + (x2 y) j + (xz) k. (8 Marks) . .. 5) C C (3.... . Session 2007/08 Sem II a) Evaluate Z C e−z ds x2 + y 2 where C is the path given by r(t) = 2 cos t i + 2 sin t j + t k for 0 ≤ t ≤ 2π........ where C is the line segment from (0............ . 0) to (2.. .. Show that F is conservative and find a scalar potential for F...... b) Given the vector field F(x..... ........ ..... . y) = ex sin y i + ex cos y j is a conservative field.. 2) C x Figure A b) Determine whether the vector function F(x. ......... 2 ......... 2).. 3 .. (6 Marks) b) A force F(x....... 1.. ....... H (5 Marks) 16...... . .. (7 Marks) c) Use Green’s theorem to evaluate C 4x2 y dx + 2y dy where C is the anticlockwise boundary of a triangular region with vertices A(0.. (7 Marks) 17. (3.. . (7 Marks) ............. . .. y............ ..... ... ......... 2) (0. If it is. B(1. .. . ... Use this potential function to find the work π done by F to move the object from the point (0.... ......... z) = (ex sin y − z sin x) i + (ex cos y) j + cos x k.................. ..... 0) to (1.... 0).... 2 (7 Marks) c) Use Green’s theorem to evaluate I [x2 + 2y + sin(x2 )] dx + [x + y + cos(y 2 )] dy C where C is the boundary of the region enclosed by the curves y = x2 and y = x in the first quadrant oriented counterclockwise.. )...... .... y).. ...... ..... ... 2) and C(0......... ...... .....SSM2083/2283: Tutorial 4 5 y........ y) = (ex sin y − y) i + (ex cos y − x − 2) j acts on an object moving in a plane. find the corresponding potential function φ(x... ........ 1 .................... ..... ......... ....... Session 2006/07 Sem I a) Evaluate the line integral Z C x dx − xy dy + yz dz.... ........... ... . 2)... 0. (i) 3 (iv) (ii) −2 3 3. (i) − 8. 0) to (1. (a) C1 ∪ C2 : 135 10. (a) 7 2 35 (vi) 48π 2 3 1 6 (b) φ = (b) φ = x2 y + xz 3 + c 1 (41 − 4π) 2 1 14. (a) 15. (7 Marks) c) Use Green’s theorem to evaluate the integral I [ex + y 2 ] dx + [ey + x2 ] dy C where C is the boundary of the region enclosed by y = x2 and y = x. −4 11.426 (vi) 2 6 1. (a) 167 2 13.8424 (c) 29 13 15 Workdone= − e−2 (b) φ = ex sin y + c (c) − 2 3 3π 1 2 (b) φ = ex sin y − xy − 2y + c Workdone= e − (c) − 16. dr is independent of the path C. 1) to (π. (i) 27 (iv) 141 1 3 6 5 (v) 1 − e3 −7 44 5 (vi) 3 6 (ii) 14 (iii) 5 4. π. 8π 1 3 1 x + xy 2 + z 2 + c 3 2 Workdone= 4 Workdone= 202 (c) − (b) φ = −ye−x + zey + c (b) φ = xey+2z + c C3 : 148 1 2 1 3 1 20 Workdone= 77. (a) −2 12. (v) 0 3 2 9. (6 Marks) ANSWERS FOR TUTORIAL 4 ´ √ 1³ √ 14 14 − 1 (iii) (ii) 2 5 6 5 (v) 29. (i) −2 (ii) 4 (iii) π (iv) 6 (v) 0 (vi) −2 1 3 (iv) 24π (ii) −4 (iii) 2 2. from (0. 1. and curve C is oriented counterclockwise. π). 0. (a) 4 30 .6 SSM2083/2283: Tutorial 4 R Show that C F . Hence find a potential function evaluate the line integral along the curve C. (i) 5 (iii) (iv) −π 5. (a) 2 3 2 6 √ 5 1 (1 − e−2π ) (b) φ = ex sin y + z cos x + c Workdone= − π (c) 17.