A) 56
B) 16
A) ln 5(9)
A)5
Quiz 22
A)
C) 23
D) 15
1
8
Example 1:
f (x) = 3
B)
C) 23
C)ln (9)
C)3
1
5
B)1.629
Math
1431
D) ln 5(5) A) 56 B) 16
LAB session 13
B) ln 9(5)
B) 1
A)1.574
C) 18
A) ln 5(9)
D)0
D)
A)5
5
8
A)
x2
1
8
Compute the upper Riemann sum for the given function f (x) = 3
to the partition P = [0; 14 ; 34 ; 1]
1
B)
D) ln 5(5)
C)ln (9)
C)3
1
5
D)1.614
D) 15
B) ln 9(5)
B) 1
C)1.589
C) 18
D)0
D)
5
8
x2 over the interval x∈[0,1] with respect
P = [0; 4 ; 34 ; 1]
f (x) = sin (x)
f (x) = sin (x)
P = [0; ⇡3 ; 2⇡
; ⇡]
3
P = [0; ⇡3 ; 2⇡
; ⇡]
3
n=4
n=4
Z
8
3x + 5 dx
0
Z
8
3x + 5 dx
0
13
13
P = [0;
Example 2:
1 3
; ; 1]
4 4
f (x) = sin (x)
Compute the lower Riemann sum for the given function
respect to the partition P = [0; ⇡ ; 2⇡ ; ⇡]
3
n=4
3
f (x) = 3
x
P = [0; 14 ; 34 ; 1]
f (x) = sin (x)
over the interval x∈ [0, ⇡]
with
P = [0; ⇡3 ; 2⇡
; ⇡]
3
n=4
Z
8
3x + 5 dx
0
Z
8
3x + 5 dx
0
13
13
A)( 1, 15 )
B
C)(
1
, 1)
5
D
A)4
B)
5
16
A)( 2.2, 0.6) [
C)( 1, 2)
D)
Example 3:
Estimate
P = [0; ⇡3 ; 2⇡
; ⇡]
3
Z
8
3x + 5 dx
by the left endpoint estimate with n = 4
0
Z
8
3x + 5 dx
0
13
13
A)
Question #
4 1 + x2
lim
2
x!1
A) 0.5 3x B) 0.25
9)
1
5
C) 18
D)
5
8
over the interval x∈[-1; 1] with respect to
P = [0; 14 ; 34 ; 1]
p
.589
D)
B)
Compute the lower Riemann sum for the given function f (x) = x2
the uniform partition of 5 points (including the endpoints)
C) 0.75
D) 1
5x 4 tan (x)
lim
x!0 2 sin (x)
10x
0
1
8
f (x) = sin (x)
P = [0; ⇡3 ; 2⇡
; ⇡]
3
n=4
D)1.614
Z
D) ln 5(5)#
Question
8
3x + 5 dx
0
What is a best way to estimate the area under the line?
A) midpoint
B) left endpoint
5
8
C) right endpoint
D) all give the same accuracy
13
Question #
Estimate the integral by the midpoint estimate with n = 3
Z
2
x + 1 dx
1
A) 1.5
B) -2.5
C) 3.75
D) 4.5
Z
x + 1 dx = 3
2
1
x +Z1 dx = 3
5
0
f(x) dx = 3
GivenZ that
7
0
f(x)Zdx7 = 2
5
Z
Z
2
Z 5 4: x + 1 dx = 3
Example
1
f(x)Zdx = 3
Find
1
f(x) dx = 2
7
0
5
f(x)Zdx7
and
Z
5
f(x) dx = 3
0
7
f(x) dx = 2
5
Z
7
f(x) dx
0
f(x) dx
0
13
13
13
f(x) dx = 2
Z
0
Z 3
Examplef(x)
5: dx = 5
Z
0
1
f(x) dx = 2
GivenZ that
5
0
f(x) dx = 4
3
Z
Find
Z
1
Z
Z
f(x) dx = 5
0
Z
0
3
f(x) dx = 5
0
5
5
9 = 7
f(x)Zdx
f(x) dx = 13
2
Z
9
2
Z
5
f(x) dx
f(x) dx = 7
5
3
Z
Z
f(x) dx =f(x)
4 dx
3
5
3
f(x) dx f(x)
= 1dx = 4
2
6
1
Z
1
Z
3
f(x) dx = 1
f(x) dx
=7
2
6
Z
6
f(x) dx = 7
f(x)
1 dx
2
Z
6
f(x) dx
2
1
f(x) dx
5
2
9
Z
Z
Z
5
f(x) dx = 4
3
Z
f(x) dx = 4
f(x) dx = 13
Z
f(x) dx = 5
0
3
9
2
1
Z
Z
and
Z
3
5
3
f(x) dx
Z
f(x) dx = 2
f(x) dx = 4
f(x) dxZ =1 13
2
Z
3
f(x)
Z 5dx
5
9
Z
Z
and
Z
0
1
Z
f(x) dx = 13
2
Z
f(x) dx = 7
Z
9
f(x) dx
5
3
f(x) dx = 4
1
3
f(x) dx = 1
2
6
f(x) dx = 7
1
Z
f(x) dx
5
9
5
2
1
Z
Z
Z
5
f(x) dx = 7
2
Z
9
f(x) dx
5
3
f(x) dx = 4
1
3
f(x) dx = 1
2
6
f(x) dx = 7
1
Z
6
f(x) dx
2
6
f(x) dx
2
14
14
14
14
Z
f(x) dx = 4
1
f(x) dx
Z
5
9
2
GivenZ that
f(x) dx = 13
5
2
Z
Z
Z
Z
1
f(x) dx
Z 9 6:
Example
5
f(x) dx
Z = 13
Find
5
3
Z
f(x) dx
Z =7
3
2
6
1
Z
5
f(x) dx
5
f(x) dx
Z 3= 4
f(x) dx = 4
1
f(x) dx
Z 3= 1
f(x) dx = 1
2
f(x) dx
Z 6= 7
f(x) dx = 7
6
2
2
Z
2
f(x) dx
Z 9
5
and
f(x) dx = 13
f(x) dx = 7
9
3
1
2
9
1
f(x) dx
Z 6
f(x) dx
Z
Z
Z
5
f(x) dx = 7
2
Z
9
f(x) dx
5
3
f(x) dx = 4
1
3
f(x) dx = 1
2
6
f(x) dx = 7
1
Z
6
f(x) dx
2
2
14
14
14
f(x) dx = 7
5
Z
3
1
2
Z
f(x) dxZ=9 4
f(x) dx
Example
Z 3 7:
5
Z 3= 1
f(x) dx
2
f(x) dx = 4
Given that
1
Z 6
Z 3= 7
f(x) dx
1
f(x) dx = 1
2
Z 6
Z dx
6
f(x)
Find
2
f(x) dx = 7
1
Z
6
f(x) dx
f(x) dx = 4
5
and
Z
Z
1
Z
3
f(x) dx = 4
1
3
f(x) dx = 1
2
6
f(x) dx = 7
1
Z
and
Z
3
f(x) dx = 1
2
6
f(x) dx = 7
1
Z
6
f(x) dx
2
6
f(x) dx
2
2
14
14
14
14
f(x)
4 6
f(x)dxdx= =
0
Z
0
Question
#
Z
Z 58
f(x) Zdx3 = 6
f(x) dx
f(x)=dx7 = 4
Given0 that
Z
5
8
and
0
Z
Zf(x)
3 dx = 7
dx = 6
Z 03 f(x)
Find 5
f(x)
dx
Z
Z
83
9
5
Z 8 f(x) dx = 4
0
f(x) dx = 7
f(x) dx
Z 55
8
Z
f(x) dx = 4
0
5
f(x) dx = 6
Z0 3
Z 8
0
5
Z
2
A) 14
f(x)
dx
2
2
Z
Z
0
f(x) dx = 9
1
B)
8
Z
C) 11
Z
0
Z0
Z
f(x) dx = 6
5
f(x) dx = 6
8
f(x) dx = 7
0
Z5
Z
3
f(x)
dx = 6
f(x) dx
7
f(x) dx = 4
5
8
Z
f(x) dx = 7
3
f(x) dx
5
5
1
7
and
f(x) dx = 4
f(x) dx = 7
Z 88
Z D)11
9 f(x) dx = 7
A) 9Z 29 B) 10
C) -9
0
f(x) dx
5 f(x) dx = 13
dx
8 13
Z 8=
Z f(x)
5
2 Z
2
3
Z 9 f(x) dx = 7
Z
=7
5
Z 5 f(x)5 dx
f(x) dx
f(x) dx = 13
8
2
f(x)
dx = 7
2Z
f(x) dx
=
3 7
Z 29
2 Z 9 Z 5 f(x) dx
Zf(x)
9 dx = 13
dx = 7
Z 9 f(x)8f(x)
dx
2
2
f(x) dx
Z 9Zdx
5 f(x)
Z 55
9
f(x)f(x)
dxdx
= 13
Z 3 f(x) dx = 7
Z 35
2
Z 5
2 f(x) dx = 4
5
f(x)
dx
=
4
ZZ 53
1Z 9
1 f(x) dx = 3
f(x) dx
dx =
= 47
Z 3 f(x) dx
1
21
5
f(x)
dx = 1
ZZ 53
Z Z3 9
Z2 5
f(x)dxdx= =
f(x)
2dx1dx
f(x)
=1
Z 6 f(x) dx = 3
f(x)
22 #
Question
2 5
1
f(x) dx = 7
ZZ 76 ZZ 6
Z1 5
5
f(x)
=
73 and
f(x)
==
9dx
dx = 2
f(x)dx
dx
7
Z f(x)
Given that
f(x)
dx
=
6
1
2
11
1
f(x) dx
Z 6
Z 72
ZZ 7 Z 5
6
f(x) dx
f(x) dx = 9
Find
f(x) dx
2f(x) dx = 2
f(x) dx
= dx
13= 6
Z f(x)
3
Z
3
9
Z8
3
f(x) dx
f(x)
dx = 13
8
2
Z
Z
9
5
2
2
f(x)dx
dx==713
f(x)
Z Z5 9
f(x)
f(x)dx
dx= 7
25
Z 3Z 9
f(x)f(x)
dx =
dx4
1
ZZ 3
5
5
f(x) dx = 1
f(x) dx = 3
2
Z 16
Z
5
f(x) dx = 7
f(x) dx = 2
1
and
Z
2Z
6
f(x) dx
7
2
f(x) dx = 9
1
Z
7
f(x) dx
2
f(x) dx
2
D) 5
7
f(x) dx
14
2
14
14
14
14
14
14
14