Transcript Document 7744813

Integration
This is not your father’s area?
The economy is so bad that the following is
happening with Snap, Crackle and Pop
A. They are thinking of replacing all
three of them with Pow
B. Kelloggs hired a “cereal” killer to
kill them all
C. Snap is spreading rumors that
“Pop was a rolling stone”
0%
D. They are selling smack, crack and
pot, respectively.
A.
0%
0%
B.
C.
0%
D.
Ask me what I should already know
The pre-requisite questions
The velocity of a body is given as v(t)=t2. Given that
6
2
t
 dt  63
3
the location of the body at t=6 is
A.
B.
C.
D.
27
54
63
Cannot be determined
0%
A.
0%
0%
B.
C.
0%
D.
The exact mean value of the function f(x) from a to b is
A.
f (a )  f (b)
2
B.
a  b
f (a )  2 f 
  f ( b)
2


4
b
C.
 f ( x)dx
a
b
D.
 f ( x)dx
a
(b  a )
0%
A.
0%
0%
B.
C.
0%
D.
Given the f(x) vs x curve, and the magnitude
b
of the areas as shown, the value of
A.
B.
C.
D.
-2
2
12
Cannot be
determined
y
5
a
 f ( x)dx
c
b
2
d
x
c
7
0%
A.
0%
0%
B.
C.
0%
D.
Given the f(x) vs x curve, and the magnitude
c
of the areas as shown, the value of
A.
B.
C.
D.
-2
2
12
Cannot be
determined
 f ( x)dx
b
y
5
a
b
2
d
x
c
7
0%
A.
0%
0%
B.
C.
0%
D.
PHYSICAL EXAMPLES
Distance covered by rocket


 m0 
  gt  dt
x   u log e 
 m0  qt 
t0 

t1


140000


x    2000 ln 
 9.8t dt

140000  2100t 

8
30
Concentration of benzene
ux

c0
 x  ut 
 x  ut 
D
c x, t   erfc
  e erfc

2 
 2 Dt 
 2 Dt 
u= velocity of ground water flow in the x -direction (m/s)
D = dispersion coefficient ( m2)
C0= initial concentration (kg/m3)
x
erfc x    e

z2
dz
Is Wal*** “short shifting” you?


a
a
P( y  a)   f ( y )dy  
Roll
1
2
3
4
5
6
7
8
9
10

1
 2
e (1 / 2)( y   ) /   dy
Number of
sheets
253
250
251
252
253
253
252
254
252
252
P( y  250)   0.3515 e
250
2
0.3881( y  252.2) 2
dy
Calculating diameter contraction
for trunnion-hub problem
T flu id
D  D  dT
Tro o m
Coefficient of Thermal Expancion
o
(in/in/ F)
7.00E-06
6.00E-06
5.00E-06
4.00E-06
3.00E-06
2.00E-06
1.00E-06
-400
-300
-200
0.00E+00
-100
0
Temperature (oF)
100
200
END
The morning after learning
trapezoidal rule
You are happy because
A. You are thinking about
your free will
B. You are delusional
C. You have to see my
pretty face for only three
more weeks
D. All of the above
0%
A.
0%
0%
B.
C.
0%
D.
Two-segment trapezoidal rule of
integration is exact for integration of
polynomials of order of at most
A.
B.
C.
D.
1
2
3
4
0%
A.
0%
0%
B.
C.
0%
D.
In trapezoidal rule, the number of
segments needed to get the exact value for
a general definite integral
A.
B.
C.
D.
1
2
1 googol
infinite
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A.
0%
0%
B.
C.
0%
D.
In trapezoidal rule, the number of points
at which function is evaluated for 8
segments is
A.
B.
C.
D.
8
9
16
17
0%
A.
0%
0%
B.
C.
0%
D.
In trapezoidal rule, the number of
function evaluations for 8 segments is
A.
B.
C.
D.
8
9
16
17
0%
A.
0%
0%
B.
C.
0%
D.
The distance covered by a rocket from t=8 to t=34 seconds
is calculated using multiple segment trapezoidal rule by
integrating a velocity function. Below is given the
estimated distance for different number of segments, n.
n
1
2
3
4
5
Value 16520 15421 15212 15138 15104
The number of significant digits at least
correct in the answer for n=5 is
A.
B.
C.
D.
1
2
3
4
0%
A.
0%
0%
B.
C.
0%
D.
The morning after learning
Gauss Quadrature Rule
Autar Kaw is looking for a stage name.
Please vote your choice.
A.
B.
C.
D.
The last mindbender
Қ (formerly known as Kaw)
Kid Cuddi
Kaw&Saki
0%
A.
0%
0%
B.
C.
0%
D.
10
 f ( x)dx
is exactly
5
1
A.
.
 f 2.5 x  7.5dx
1
1
B.
2.5  f (2.5 x  7.5)dx
1
1
C.
5  f (5 x  5)dx
1
1
D.
5  (2.5 x  7.5) f ( x)dx
1
0%
A.
0%
0%
B.
C.
0%
D.
A scientist would derive one-point Gauss
Quadrature Rule based on getting exact results of
integration for function f(x)=a0+a1x.
The one-point
b
rule approximation for the integral  f ( x)dx is
A.
.ba
2
B.
C.
D.
a
[ f (a )  f (b)]
ab
(b  a ) f (
)
2
b  a   b  a  1  b  a 
 
 
 f 
2   2  3
2 
 b  a  1  b  a 

f 
 
2 
 2  3
(b  a) f (a)
0%
A.
0%
0%
B.
C.
0%
D.
For integrating any first order
polynomial, the one-point Gauss
quadrature rule will give the same results
as
A.
B.
C.
D.
1-segment trapezoidal rule
2-segment trapezoidal rule
3-segment trapezoidal rule
All of the above
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A.
0%
0%
B.
C.
0%
D.
A scientist can derive a one-point quadrature rule
for integrating definite integrals based on getting
exact results of integration for the following
function
A.
B.
C.
D.
a0+a1x+a2x2
a1x+a2x2
a1x
a2x2
0%
A.
0%
0%
B.
C.
0%
D.
For integrating any third order
polynomial, the two-point Gauss
quadrature rule will give the same results
as
A.
B.
C.
D.
1-segment trapezoidal rule
2-segment trapezoidal rule
3-segment trapezoidal rule
None of the above
0%
A.
0%
0%
B.
C.
0%
D.
The highest order of polynomial for which
the n-point Gauss-quadrature rule would
give an exact integral is
A.
B.
C.
D.
n
n+1
2n-1
2n
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A.
0%
B.
0%
C.
0%
D.
END
Ab scientist an approximate formula for integration as
 f ( x)dx  c f ( x )
1
1
,where a  x1  b
a
The values of c1 and x1 are found by assuming that the formula is exact
for the functions of the form a0x+a1x2 polynomial. Then the resulting
formula would be exact for integration.
A.
. b  a [ f (a )  f (b)]
2
B.
C.
D.
(b  a ) f (
ab
)
2
b  a   b  a  1  b  a 
 
 
 f 
2   2  3
2 
 b  a  1  b  a 

f 
 
2 
 2  3
(b  a) f (a)
0%
A.
0%
0%
B.
C.
0%
D.
2.2
The exact value of
x
xe
 dx most nearly is
0.2
A.
B.
C.
D.
7.8036
11.807
14.034
19.611
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A.
0%
0%
B.
C.
0%
D.
The area of a circle of radius a can be found by the
following integral
 a
a
A.
)
2

 x 2 dx
0
2
B.

) a 2  x 2 dx
0
a
C.
4)  a 2  x 2 dx
0
D.
a

)
a 2  x 2 dx
0
0%
A.
0%
0%
B.
C.
0%
D.
9
2
x
The value of  dx by using one segment
5
trapezoidal rule is most nearly
A.
B.
C.
D.
201.33
212.00
424.00
742.00
0%
A.
0%
0%
B.
C.
0%
D.
The velocity vs time is given below. A good estimate
of the distance in meters covered by the body
between t=0.5 and 1.2 seconds is
t(s)
0
0.5
1.2
1.5
1.8
v(m/s)
0
213
256
275
300
A.
B.
C.
D.
213*0.7
256*0.7
256*1.2-213*0.5
½*(213+256)*0.7
0%
A.
0%
0%
B.
C.
0%
D.
Velocity distribution of a fluid flow through
a pipe varies along the radius, and is given
by v(r). The flow rate through the pipe of
radius a is given by
A.
v(a)a
B.
v(0)  v(a) 2

a
2
C.
.
a
 v(r )dr
0
D.
2
a
2  v(r )rdr
0
0%
A.
0%
0%
B.
C.
0%
D.
You are asked to estimate the water flow rate in a pipe of radius 2m at a remote
area location with a harsh environment. You already know that velocity varies
along the radial location, but do not know how it varies. The flow rate, Q is
2
given by
Q   2rVdr
To save money,0 you are allowed to put only two velocity probes (these probes
send the data to the central office in New York, NY via satellite) in the pipe.
Radial location, r is measured from the center of the pipe, that is r=0 is the
center of the pipe and r=2m is the pipe radius. The radial locations you would
suggest for the two velocity probes for the most accurate calculation of the flow
rate are
A. 0,2
B. 1,2
C. 0,1
D. 0.42,1.58
0%
A.
0%
0%
B.
C.
0%
D.
Given the f(x) vs x curve, and the magnitude of the
areas as shown, the value of
A.
B.
C.
D.
5
y
12
14
Cannot be determined
a
 f ( x)dx
0
5
a
b
2
c
x
7
0%
A.
0%
0%
B.
C.
0%
D.
Given the f(x) vs x curve, and the magnitude
b
of the areas as shown, the value of
A.
B.
C.
D.
-7
-2
12
Cannot be
determined
 f ( x)dx
0
y
5
a
b
2
c
x
7
0%
A.
0%
0%
B.
C.
0%
D.
Given the f(x) vs x curve, and the magnitude
b
of the areas as shown, the value of f ( x)dx

A.
B.
C.
D.
-7
-2
7
12
a
y
5
a
b
2
c
x
7
0%
A.
0%
0%
B.
C.
0%
D.
The value of the integral
A. x3
B. x3 +C
C. x3/3
D. x3/3 +C
E. 2x
2
x
 dx
0%
0%
A.
B.
0%
0%
C.
D.
0%
E.
b
Physically, integrating  f ( x)dx means finding the
a
A.
B.
C.
D.
Area under the curve from a to b
Area to the left of point a
Area to the right of point b
Area above the curve from a to b
0%
A.
0%
0%
B.
C.
0%
D.
The velocity of a body is given as v(t)=t2. Given that
6
2
t
 dt  63
3
the distance covered by the body between t=3 and t=6 is
A.
B.
C.
D.
27
54
63
Cannot be determined
0%
A.
0%
B.
0%
C.
0%
D.