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Past Years Compilation (mid sem and final)
CHAPTER 4 (SEF 1134)
SEM 1, 2008/2009
1 - x - 1  x  0
 2
0 x2
1. Given f ( x)   x
- 1
2 x3

Evaluate
Ans:

3
f ( x) dx .
1
[5]
25
6
π
4
4
∑sink ( )
2. Find the exact value
Ans:
k =1
[3]
3
( 2  1)
4
3. By using definition of area under the curve y = f(x) as
n
A = lim ∑f (x *k )Δx
n→∞k =1
with
xk*
as the right endpoint of each subinterval,
show that the area under the the curve y=mx over interval [a,b] is
a+b
m (b - a) (
)
2
4. Evaluate :

x5
2
x3  1
1
dx
[7]
Ans: 4.313
5. Evaluate :
Ans:

e2x
dx
1  e4x
[4]
1
tan -1 (e 2 x )  C
2
3
2
i=1
j=1
6. Evaluate : ∑ [ ∑(i + j)]
[4]
Ans:21
1
7. Evaluate the integrals by applying an appropriate formula from geometry
0

 x 2  8 x dx
[4]
8
Ans : 8 π
SEM 2, 2008/2009
5
8. If

f x dx  12 and
1
5

f  x dx  36 , find
4
4
 f x dx  36 .
[3]
1
9. Evaluate the integral:

 2 cos x  cos x dx
[5]
1  x 2 dx  12 by interpreting it in terms of areas.
[4]
1
0
1
10. Evaluate
x
0
11. Show by using definition that the area under the curve y  x  2x 2 over [0,5] is
25
23
. Use right endpoint approximation.
6
[10]
12. Find the integrals:
 1  x dx
b)  sin 1  x 2 cos1  x  dx
a)
[5]
4
[4]
13. Use geometry formula to find the net signed area between the curve
y = 1 3 x and the interval
ans: 
1
3
[-1, ].
[4]
1
3
50
14. By using properties of sigma evaluate
 (k  1)
2
[5]
k 1
Ans: 45525
15. Express the following function in a closed form and find the limit.
12  22  32  ...  n 2
lim
n
n3
[5]
Ans: 2
16. Use Midpoint Approximation with n = 6 to approximate

8
2
1
dx [5]
1 x2
2
Ans: 0.2907
SEM 3, 2008/2009
3x 2 ; x  2
17. Given f ( x)  
, evaluate
4 ; x  2
5
 f ( x) dx .
[3]
2
Ans: 19
18. Evaluate the following integral using an appropriate formula from geometry.
5
x
25  x 4 dx .
[4]
0
1

Ans:   ( 25) 
8

19. Use Definition with x k as the left endpoint of each subinterval to find the area under the curve
f ( x)  1  x 2 over [0,3].
Ans: -6
20. Evaluate
Ans:

x x  3 dx
5
3
2
x  3 2  2x  3 2  C
5
1

21. Evaluate
sin 1 x
2
1 x 2
0
Ans:
[4]
[5]
dx
2
8
1
2
22. Evaluate
x
 1  4x
4
dx
[5]
0
Ans:

16
20
23. Evaluate :
 [k
2
 3]
[3]
k 1
Ans: 2930
24. By using the definition of area, use the right endpoint estimation to find the area under the
curve 𝑦 = 4𝑥 3 + 3𝑥 2 on the interval [0,1].
[6]
Ans: 2
𝜋
25. ∫04 √1 + cos 4𝑥 𝑑𝑥
(Hint : Use trigonometric identity)
[5]
3
Ans:
√2
2
26. Evaluate : ∫
Ans: 
2𝑥+1
5 𝑑𝑥
(𝑥+1) ⁄2
4

x 1
[4]
2
3x  1
3
2
C
SEM 1, 2009/2010
27. Express 1 + 2 + 22 + 23 + 24 + 25 + 26 in sigma notation with k = 3 as the lower limit.
[3]

2
28. Evaluate :
1
 cos x dx
2

0
Ans:
3 1
[5]

12

n 
n
 f ( x )

29. By using the definition of an area, A  lim 
k 1
k
x

 with 𝑥𝑘 as the right-end point, show

that the triangle with vertices (0, 0), (0, h) and (b, 0) has an area of
𝟏
𝒃𝒉
𝟐
.
Refer to figure 1 below.
h
b
Equation of line :
2

1
Ans:
155
4
31. Evaluate :
x y
 1
b h
1

 x  2 x  dx
4
30. Evaluate :
Figure 1


[4]

e tan 2 1  tan 2 2 d
[4]
4
Ans :
32.
ex
 4  3e2 x dx
Ans:
33.
1 tan 2
e
C
2
1
2 3
tan
1
x
x4  3
Ans:
1
2 3
[4]
3e x
C
2
1
[5]
dx
x2
sec 1
3
C
2 ; x  2
x ; x  2
34. Given f ( x)  
3
By using the geometry formula, evaluate

[3]
f ( x ) dx
1
Ans:
17
2
35. Express the following sum in closed form and find the limit :
[5]
n
 2 2k k 2 
lim    2  3 
n 
n
n 
k 1  n
Ans:
36.

10
3
3



x sin 1  x 2  dx


Ans: 
[4]
3


2
2
cos u  C   cos1  x 2   C
3
3


SEM 2, 2009/2010
37. Evaluate :
∑𝟑𝒌=𝟎 (−𝟐)𝒌 𝒔𝒊𝒏 𝒌𝝅𝟔
[3]
Ans : −9 + 2√3
5
|𝑥 − 2| 𝑖𝑓 𝑥 ≥ 0
38. Given 𝑓(𝑥) = {
𝑥 + 2 𝑖𝑓 𝑥 < 0
[5]
6
Find ∫−4 𝑓(𝑥) 𝑑𝑥.
Ans : 10
39. Evaluate the integral by completing the squares and applying appropriate formula from
geometry 
Ans:
3
0
6 x  x 2 dx
[4]
9
4

n 
the area under the curve 𝑓(𝑥) = 6𝑥 − 𝑥
Ans: 18
k 1
2

ln 2
Ans:  
k
x
over [ 0,3]
ln(2 / 3 )
41. Evaluate the following integrals :

n
 f ( x )  with 𝑥𝑘 as the right-end point, find

40. By using the definition of an area, A  lim 
e x
1  e 2 x
dx
[4]

6
SEM 3, 2009/2010
42. Evaluate
Ans :
1
2 x  x 2 dx
[6]
5
3
1
43. Find

3
7
 (3i  j)
[4]
j 0 i 5
Ans : 105
n
44. Use Definition lim
n
 f (x
k 1
*
k
) of area under the curve with xk as the right endpoint of each
subinterval to find the area under the curve f ( x )  x 3  6 x over [0,3].
[10]
Ans : -6.75
45. Evaluate

cos(sec 1 x )
x x 2 1
dx
[3]
Ans : sin (sec-1x) + c
6
EXTRE QUESTIONS
46. Evaluate
e
e2
47. Evaluate

e
48. Evaluate
2x
1  e x dx
dx
x ln x
dx
 x4  ln x  
2
7