(a) (6 marks) Determine the values of A, B and C such that 1 x(x 2 + 1) = A x + Bx + C x 2 + 1 . Hence evaluate Z 1 x(x 2 + 1)dx. (b) (6 marks) Find the area under the curve y = xe−x for x ≥ 0.
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(a) (6 marks) Determine the values of A, B and C such that 1 x(x
2...
ABC 6. (a) Find constants A, B and C such that (3 marks) (b) Hence determine...
ABC 6. (a) Find constants A, B and C such that (3 marks) (b) Hence determine dx. r(1-r2) (2 marks) (c) Evaluate J2 2 marks) dx. Write your answer as a single log. 1-A)
14 1. (10 marks) Let C be the curve 27x - y = 0 between y=0...
14 1. (10 marks) Let C be the curve 27x - y = 0 between y=0 and y=4. Sketch the graph of this curve. In each part, set up, but do not ovaluate, an integral or a sum of integrals that solves the problem. (a) Find the area of the surface generated by revolving C about the x-axis by integrating with respect to x (6) Find the area of the surface generated by revolving C about the y-axis by integrating...
Z=61 Task 3: Answer the following: a. Evaluate: Siz cos(x) sin?(x) dx (10 Marks) b. The...
Z=61
Task 3: Answer the following: a. Evaluate: Siz cos(x) sin?(x) dx (10 Marks) b. The moment of inertia, I, of a rod of mass 'm' and length 4r is given by Ar (2mx? dx where 'x' is the distance from an axis of rotation. Find I. (5 Marks) 2r Task 4: Answer the following: Using the Trapezoidal rule, find the approximate the area bounded by the curve y = ze), the x-axis and coordinates x = 0, x =...
6.a) Let -2 5 -6 10 ), 2) an Evaluate the matrix 8 marks) b) Find the area enclosed between the following curves and sketch the region. 5x2-2. f(x)--3x2 + 30 and g(x) marks) c) Evaluate the follow...
6.a) Let -2 5 -6 10 ), 2) an Evaluate the matrix 8 marks) b) Find the area enclosed between the following curves and sketch the region. 5x2-2. f(x)--3x2 + 30 and g(x) marks) c) Evaluate the following integrals. Give your answers to 2 decimal places. dac i) 2 (3+2 dx. (8 marks) [25 marks]
6.a) Let -2 5 -6 10 ), 2) an Evaluate the matrix 8 marks) b) Find the area enclosed between the following curves and sketch...
Question 6 [10 marks a) Let f(x) = x for each xe [a,b]. Show that for...
Question 6 [10 marks a) Let f(x) = x for each xe [a,b]. Show that for any number of subintervals, the global error js(x)dx-SUS J) = 0. [6] Hint: Obtain the local error first and then calculate the global error. SCS ,h) denotes approximation using the composite Simpson's Rule. b) Determine the minimum number of subintervals so that the upper bound of the (absolute) global error for the composite Simpson's Rule applied to ja?-10x”) dx is less than 10%. [41...
1. (6 marks) Find the volume of the solid enclosed by the paraboloid 2 = 1...
1. (6 marks) Find the volume of the solid enclosed by the paraboloid 2 = 1 - 22 - y2 and the coordinate planes of the first octant O = {(x, y, z) | x > 0, y > 0, z>0}. 2. (7 marks) Calculate SS/ (82 +93) dr dy dz. where E is the upper hemisphere x2 + y2 + 22 < 1 and 2 > 0. 3. (7 marks) Evaluate the integral SL (x + y) er?-y dA...
4. (a Let (sin( x cos( ) dr + (x cos(x + y) - 2) dy....
4. (a Let (sin( x cos( ) dr + (x cos(x + y) - 2) dy. dz= Show that dz is an exact differential and determine the corresponding function f(x,y) Hence solve the differential equation = z sin( Cos( y) 2 x cos( y) dy 10] (b) Find the solution of the differential equation d2y dy 2 y e dx dæ2 initial conditions th that satisfi 1 (0) [15] and y(0) 0
4. (a Let (sin( x cos( ) dr...
All of 10 questions, please. 1. Find and classify all the critical points of the function. f(x,y) - x2(y - 2) - y2 » 2. Evaluate the integral. 3. Determine the volume of the solid that is inside the...
All of 10 questions, please.
1. Find and classify all the critical points of the function. f(x,y) - x2(y - 2) - y2 » 2. Evaluate the integral. 3. Determine the volume of the solid that is inside the cylinder x2 + y2- 16 below z-2x2 + 2y2 and above the xy - plane. 4. Determine the surface area of the portion of 2x + 3y + 6z - 9 that is in the 1st octant. » 5. Evaluate JSxz...
(a) Evaluate: L (2+5x)e 'dx 9Marks (b) Determine the area enclosed between the curves y =...
(a) Evaluate: L (2+5x)e 'dx 9Marks (b) Determine the area enclosed between the curves y = x + 1 and y = 7-X 8Marks (C) Water is leaking from a pipe and the volume. V litres, of the water is decreasing at the rate (6-21) litres per minutes after t minutes. Assuming that V = 9 when t = 0, find V in terms of t. 7 Marks (d) Find the equation of the curve that passes through (26) If...
(a) Evaluate: L (2+5x)e 'dx 9Marks (b) Determine the area enclosed between the curves y =...
(a) Evaluate: L (2+5x)e 'dx 9Marks (b) Determine the area enclosed between the curves y = x + 1 and y = 7-X 8Marks (C) Water is leaking from a pipe and the volume. V litres, of the water is decreasing at the rate (6-21) litres per minutes after t minutes. Assuming that V = 9 when t = 0, find V in terms of t. 7 Marks (d) Find the equation of the curve that passes through (26) If...
ABC 6. (a) Find constants A, B and C such that (3 marks) (b) Hence determine dx. r(1-r2) (2 marks) (c) Evaluate J2 2 marks) dx. Write your answer as a single log. 1-A)
14 1. (10 marks) Let C be the curve 27x - y = 0 between y=0 and y=4. Sketch the graph of this curve. In each part, set up, but do not ovaluate, an integral or a sum of integrals that solves the problem. (a) Find the area of the surface generated by revolving C about the x-axis by integrating with respect to x (6) Find the area of the surface generated by revolving C about the y-axis by integrating...
Z=61
Task 3: Answer the following: a. Evaluate: Siz cos(x) sin?(x) dx (10 Marks) b. The moment of inertia, I, of a rod of mass 'm' and length 4r is given by Ar (2mx? dx where 'x' is the distance from an axis of rotation. Find I. (5 Marks) 2r Task 4: Answer the following: Using the Trapezoidal rule, find the approximate the area bounded by the curve y = ze), the x-axis and coordinates x = 0, x =...
6.a) Let -2 5 -6 10 ), 2) an Evaluate the matrix 8 marks) b) Find the area enclosed between the following curves and sketch the region. 5x2-2. f(x)--3x2 + 30 and g(x) marks) c) Evaluate the following integrals. Give your answers to 2 decimal places. dac i) 2 (3+2 dx. (8 marks) [25 marks]
6.a) Let -2 5 -6 10 ), 2) an Evaluate the matrix 8 marks) b) Find the area enclosed between the following curves and sketch...
Question 6 [10 marks a) Let f(x) = x for each xe [a,b]. Show that for any number of subintervals, the global error js(x)dx-SUS J) = 0. [6] Hint: Obtain the local error first and then calculate the global error. SCS ,h) denotes approximation using the composite Simpson's Rule. b) Determine the minimum number of subintervals so that the upper bound of the (absolute) global error for the composite Simpson's Rule applied to ja?-10x”) dx is less than 10%. [41...
1. (6 marks) Find the volume of the solid enclosed by the paraboloid 2 = 1 - 22 - y2 and the coordinate planes of the first octant O = {(x, y, z) | x > 0, y > 0, z>0}. 2. (7 marks) Calculate SS/ (82 +93) dr dy dz. where E is the upper hemisphere x2 + y2 + 22 < 1 and 2 > 0. 3. (7 marks) Evaluate the integral SL (x + y) er?-y dA...
4. (a Let (sin( x cos( ) dr + (x cos(x + y) - 2) dy. dz= Show that dz is an exact differential and determine the corresponding function f(x,y) Hence solve the differential equation = z sin( Cos( y) 2 x cos( y) dy 10] (b) Find the solution of the differential equation d2y dy 2 y e dx dæ2 initial conditions th that satisfi 1 (0) [15] and y(0) 0
4. (a Let (sin( x cos( ) dr...
All of 10 questions, please.
1. Find and classify all the critical points of the function. f(x,y) - x2(y - 2) - y2 » 2. Evaluate the integral. 3. Determine the volume of the solid that is inside the cylinder x2 + y2- 16 below z-2x2 + 2y2 and above the xy - plane. 4. Determine the surface area of the portion of 2x + 3y + 6z - 9 that is in the 1st octant. » 5. Evaluate JSxz...
(a) Evaluate: L (2+5x)e 'dx 9Marks (b) Determine the area enclosed between the curves y = x + 1 and y = 7-X 8Marks (C) Water is leaking from a pipe and the volume. V litres, of the water is decreasing at the rate (6-21) litres per minutes after t minutes. Assuming that V = 9 when t = 0, find V in terms of t. 7 Marks (d) Find the equation of the curve that passes through (26) If...
(a) Evaluate: L (2+5x)e 'dx 9Marks (b) Determine the area enclosed between the curves y = x + 1 and y = 7-X 8Marks (C) Water is leaking from a pipe and the volume. V litres, of the water is decreasing at the rate (6-21) litres per minutes after t minutes. Assuming that V = 9 when t = 0, find V in terms of t. 7 Marks (d) Find the equation of the curve that passes through (26) If...
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