Solution11.pdf - HOMEWORK XI Calculus I Solution 1. Express...

HOMEWORK XI
Calculus I Solution
1. Express the limit as a definite integral on the interval [2 , 6]: lim
n → + ∞
n X
i =1
cos x *
i
x *
i
Δ x
i
Solution :
R
6
2
cos x
x
dx
2. Use the Midpoint Rule with n = 5 to estimate the integral
R
2
0
x
x +1
dx . Round
the answer to four decimal places.
3. Evaluate the integrals:
(i).
R
3
- 2
( x
2
- 3) dx
(iii).
R
1
1

HOMEWORK XI
Calculus I Solution
Z
1
- 1
e
x +1
dx =
Z
1
- 1
e
x
e dx
= e
Z
1
- 1
e
x
dx
= e e
x
|
1
- 1
= e ( e -
1
e
)
= e
2
- 1
(iv).
R
3 π
2
0
| sin x | dx
Solution :
Z
3 π
2
0
| sin x | dx =
Z
π
2
0
| sin x | dx +
Z
3 π
2
π
| sin x | dx
=
Z
π
0
sin x dx +
Z
3 π
2
π
- sin x dx
= - cos x |
π
0
+ cos x |
3 π
2
π
= 2 + 1
= 3
4. Find the general indefinite integral
R
sin 2 x
sin x
dx
5. The velocity of a moving particle is v ( t ) = t 2 - 2 t - 8.
2

HOMEWORK XI