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

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

HOMEWORK XI
Calculus I
4. Find the general indefinite integral
R
sin 2 x
sin x
dx
2

HOMEWORK XI
Calculus I Solution
(i). Find the displacement during the time [1 , 6].
Solution :
Z
6
1
v ( t ) dt =
Z
6
1
t
2
- 2 t - 8 dt
=
t 3
3
- t
2
- 8 t
6
1
= -
10
3
(ii). Find the distance traveled by the particle during time [1 , 6].
Solution :
v ( t ) = t 2 - 2 t - 8 = ( t + 2)( t - 4), so v ( t ) > 0 when t < - 2 or t > 4, v ( t ) <
when - 2 < t < 4.
Z
6
1
| v ( t ) | dt =
Z
4
1
| v ( t ) | dt +
Z
6
4
| v ( t ) | dt
=
Z
4
1
- v ( t ) dt +
Z
6
4
v ( t ) dt
=
Z
4
1
- t
2
+ 2 t + 8 dt +
Z
6
4
t
2
- 2 t - 8 dt
= ( -
t
3
3
+ t
2
+ 8 t )
4
1
+ (
t
3
3
- t
2
- 8 t )
6
4
=
98
3
3