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Integration
This is not your father’s area?
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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”
D. They are selling smack,
crack and pot, respectively.
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A.
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B.
C.
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D.
Ask me what I should already know
The pre-requisite questions
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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
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B.
C.
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A.
D.
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
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C.
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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
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B.
C.
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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)
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A.
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B.
C.
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D.
Given the f(x) vs x curve, and the magnitude of
b
the areas as shown, the value of
f ( x )dx
0
y
5
a
b
2
c
x
7
A.
B.
C.
D.
-7
-2
12
Cannot be determined
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B.
C.
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D.
Given the f(x) vs x curve, and the magnitude of
b
the areas as shown, the value of
f ( x )dx
a
y
5
a
b
2
c
x
7
A.
B.
C.
D.
-7
-2
7
12
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C.
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PHYSICAL EXAMPLES
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Distance Covered By Rocket
m0
gtdt
x u loge
m0 qt
t0
t1
140000
x 2000ln
9.8t dt
140000 2100t
8
30
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Concentration of Benzene
c0
c x, t
2
ux
x ut
x ut
D
e erfc
erfc
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
erfcx e
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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
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2
0.3881( y 252.2 ) 2
dy
Calculating diameter contraction
for trunnion-hub problem
T fluid
D D dT
Troom
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
0.00E+00
-100
0
-200
o
Temperature ( F)
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100
200
END
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7.02
Trapezoidal Rule
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The morning after learning
trapezoidal rule…
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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
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A.
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C.
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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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B.
C.
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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%
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A.
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B.
C.
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D.
In trapezoidal rule, the number of function
evaluations for 8 segments is
A.
B.
C.
D.
8
9
16
17
0%
A.
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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
Value
2
3
4
16520 15421 15212 15138
5
15104
The number of significant digits at least
correct in the answer for n=5 is
A.
B.
C.
D.
1
2
3
4
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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
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7.05
Gauss Quadrature Rule
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The morning after learning
Gauss Quadrature Rule…
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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
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10
f ( x )dx
is exactly
5
1
A.
f 2.5x 7.5dx
1
1
B.
2.5 f (2.5x 7.5)dx
1
1
C.
5 f (5x 5)dx
1
1
D.
5 (2.5x 7.5) f ( x)dx
1
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A scientist would derive one-point Gauss Quadrature
Rule based on getting exact results of integration for
function f(x)=a0+a
1x. The one-point rule approximation
b
for the integral f ( x ) dx is
a
ba
[ f (a ) f (b)]
2
ab
B.
(b a ) f (
)
2
b a 1 b a
f
C.
2
ba 2 3
2 ba 1 ba
f
2
2 3
A.
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(b a) f (a)
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5.
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 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
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B.
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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
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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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END
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