Integration

This is not your father’s area?

Ask me what I should already know

The pre-requisite questions

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

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.

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

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

PHYSICAL EXAMPLES

Distance covered by rocket

1

0 0

0elog

t

t

dtgtqtm

mux

30

8

8.92100140000

140000ln2000 dtt

tx

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)

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

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

END

The morning after learning trapezoidal rule

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

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

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

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

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

The morning after learning Gauss Quadrature Rule

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

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

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

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

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

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

END