# MATHEMATICS , EXAM 08A INTEGRATION QUESTION 1 Determine the by lindahy

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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.

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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

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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).}

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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
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

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
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.

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