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Integration
This is not your father’s area?
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Ask me what I should already know
The pre-requisite questions
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The velocity of a body is given as v(t)=t2. Given that
A. B. C. D.
0% 0%0%0%
the location of the body at t=6 is
6
3
2 63dtt
A. 27B. 54C. 63D. Cannot be determined
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The exact mean value of the function f(x) from a to b is
A. B. C. D.
0% 0%0%0%
2
)()( bfaf
4
)(2
2)( bfba
faf
b
a
dxxf )(
)(
)(
ab
dxxfb
a
A.
B.
C.
D.
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Given the f(x) vs x curve, and the
magnitude of the areas as shown, the
value of
A. B. C. D.
0% 0%0%0%
b
c
dxxf )(y
xa5
7
2b dc
A. -2B. 2C. 12D. Cannot be
determined
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Given the f(x) vs x curve, and the
magnitude of the areas as shown, the
value of
A. B. C. D.
0% 0%0%0%
c
b
dxxf )(
y
xa5
7
2b dc
A. -2B. 2C. 12D. Cannot be
determined
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PHYSICAL EXAMPLES
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Distance covered by rocket
1
0 0
0elog
t
t
dtgtqtm
mux
30
8
8.92100140000
140000ln2000 dtt
tx
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Concentration of benzene
dzexerfcx
z
2
Dt
utxerfce
Dt
utxerfc
ctxc D
ux
222, 0
u= velocity of ground water flow in the x -direction (m/s)D = dispersion coefficient (m2)C0= initial concentration (kg/m3)
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Is Wal*** “short shifting” you?
a
y
adyedyyfayP
2/)()2/1(
2
1)()(
Roll Number of sheets
1 2532 2503 2514 2525 2536 2537 2528 2549 25210 252
250
)2.252(3881.0 2
3515.0)250( dyeyP y
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Calculating diameter contraction for trunnion-hub problem
0.00E+00
1.00E-06
2.00E-06
3.00E-06
4.00E-06
5.00E-06
6.00E-06
7.00E-06
-400 -300 -200 -100 0 100 200
Temperature (oF)
Co
effi
cien
t o
f T
her
mal
Exp
anci
on
(in
/in
/oF
)
fluid
room
T
T
dTDD
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END
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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.
0% 0%0%0%
A. 1B. 2C. 3D. 4
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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.
0% 0%0%0%
A. 1B. 2C. 1 googolD. infinite
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In trapezoidal rule, the number of points at which function is evaluated for 8 segments is
A. B. C. D.
0% 0%0%0%
A. 8B. 9C. 16D. 17
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In trapezoidal rule, the number of function evaluations for 8 segments is
A. B. C. D.
0% 0%0%0%
A. 8B. 9C. 16D. 17
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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.
A. B. C. D.
0% 0%0%0%
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. 1B. 2 C. 3 D. 4
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The morning after learning Gauss Quadrature Rule
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A. B. C. D.
0% 0%0%0%
is exactly
A. .
B.
C.
D.
1
1
5.75.2 dxxf
1
1
)5.75.2(5.2 dxxf
1
1
)55(5 dxxf
1
1
)()5.75.2(5 dxxfx
10
5
)( dxxf
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A. B. C. D.
0% 0%0%0%
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 rule approximation for the integralA. .
B.
C.
D.
)]()([2
bfafab
)()( afab
)2
()(ba
fab
23
1
223
1
22
ababf
ababf
ab
b
a
dxxf )( is
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For integrating any first order polynomial, the one-point Gauss quadrature rule will give the same results as
A. B. C. D.
0% 0%0%0%
A. 1-segment trapezoidal ruleB. 2-segment trapezoidal ruleC. 3-segment trapezoidal ruleD. All of the above
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A. B. C. D.
0% 0%0%0%
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. a0+a1x+a2x2
B. a1x+a2x2
C. a1x
D. a2x2
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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.
0% 0%0%0%
A. 1-segment trapezoidal ruleB. 2-segment trapezoidal ruleC. 3-segment trapezoidal ruleD. 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.
0% 0%0%0%
A. nB. n+1C. 2n-1D. 2n
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END