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MATHEMATICS , EXAM 08A INTEGRATION QUESTION 1 Determine the following integration R 1 + x2 √ dx x QUESTION 2 Perform the following integration R dx (5x + 8)3 QUESTION 3 Obtain the value of the following integral R 2 3 − 2x 1 dx x4 QUESTION 4 The sketch shows the parabola x = 3 + 2y − y 2 . The shaded area is given by R3 2 0 (3 + 2y − y )dy; determine the magnitude of this area. QUESTION 5 The sketch shows the curve y = x1/2 (1 − x). Find the area enclosed between the curve and the x− axis. 1 QUESTION 6 Show that x3 −3x2 +2x = x(x−1)(x −2). Use this to sketch the curve y = x3 −3x2 +2x. QUESTION 7 Find the area enclosed under the curve y = (x − 3)5 between x = 3 and x = 4. QUESTION 8 The arc of the parabola y = x2 between the points O(0, 0) and B(2, 4) is rotated about the y− axis. Find the volume of the paraboloid of revolution so formed. Determine the volume of the slice of this paraboloid between the planes y = 1, y = 4 QUESTION 9 Find the volume of the solid generated when the region bounded by the curve y = x2 − 2 and the x− axis is rotated about the y− axis. QUESTION 10 On the same diagram sketch the curve y = x3 and the line y = x. (i) Find the area enclosed between the curve and the line in the positive quadrant. (ii) This area is rotated about the x− axis. Find the volume of the solid so formed. QUESTION 11 2 The curve y = x2 , y 2 = x are shown on the sketch. Show that A has coordinates (1, 1). (i) Calculate the area bounded by the given curves. (ii) Determine the volume of the solid generated when this area is rotated about the x− axis. QUESTION 12 Use the Trapezoidal Rule to ﬁnd an approximation to the integral R 40 ◦ 0 sin x dx; three strips Give answer correct to 4 signiﬁcant ﬁgures. QUESTION 13 x 0 0.2 0.4 0.6 0.8 1 y 1.7 2.6 3.2 4.1 4.9 5.8 Use the table given above, and The Trapezoidal Rule, to calculate the approximate value of R1 0 ydx QUESTION 14 Test the accuracy of the Trapezoidal approximation to the deﬁnite integral given R4√ 0 xdx; n = 4 {Note this involves obtaining an approximation and an exact answer.} QUESTION 15 Use Simpson’s Rule and the following table to calculate the approximate value of R2 1 f (t)dt t 1 1.5 2 f (t) 0 0.406 0.692 QUESTION 16 Use Simpson’s Rule with three points (i.e. two strips) to approximately evaluate the deﬁnite integral R6 x 0 2 dx {Compare the result with the previous set n. 3 where the Trapezoidal Rule was used with four points (i.e. three intervals).} 3 QUESTION 17 The Simpson’s Rule to ﬁnd an approximate value of the following deﬁnite integral, taking n the number of intervals as stated. R 1 −x2 0 e dx; n = 2 QUESTION 18 R Find the approximate value of 090 cos x0 dx, correct to 5 signiﬁcant ﬁgures, taking 6 strips, by (a) The Trapezoidal Rule (b) Simpson’s Rule (The exact value is 180/π + 57.29578 . . . . . . ) QUESTION 19 R 1 Calculate the approximate value of −1 log10 (x + 2)dx, using 4 strips, by (a) The Trapezoidal Rule (b) Simpson’s Rule (Give answers correct to 4 decimal places.) QUESTION 20 √ Show that the equation y = 16 − x2 represents a semi-circle, centre the origin O with radius 4 units. √ Find an approximation for the area enclosed between the curve y = 16 − x2 , the x− axis and the ordinates x = −4, x = 4 using 8 sub-intervals by means of Simpson’s Rule and also the Trapezoidal Rule. (Give results correct to 5 signiﬁcant ﬁgures. ) How do these approximations compare with the area obtained from the formula A = 1 πr2 ? 2 ANSWERS 1/2 2 5/2 (5x + 8)−2 1. 2x + x = C 2. − + C 3. 1/8 4. 9 unit2 5. 4/15 unit2 5 10 6. 1/2 units2 7. 1/6 units2 8. 8π, 15π/2 9. 2π 10. (i) 1/4 (ii) 4π/21 11. (i) 1/3 (ii) 3π/10 12 12.5 13. 3.71 14. 5.146 15. 0.386 16. 97 17. 0.7472 18. (a) 56.968 (b) 0.5625 19. (a) 0.5568 (b) 57.297 20. 24.449; 23.966; exact 8π + 25.133 NB. Suggestions 1. Before your try the exam please read and learn the theory involved on this chapter 2. Solve this exam without textbook and be fear. 3. Time spent = 2 hours 4. Evaluation : Solve 14 questions = 70 %; 18 questions = 80 %; 27 questions = 90 % All questions have equal value. 4