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Integration Methods: Trapezoidal Rule, Gaussian Quadrature, and Change of Variables - Prof, Quizzes of Mechanical Engineering

University of South Florida (USF)Mechanical Engineering
Prof. Autar Kaw

Sample quiz questions for the eml3041 – computational methods course focusing on integration techniques. The questions cover various methods such as the trapezoidal rule, gaussian quadrature, and change of variables. Students are required to find the value of integrals using these methods and identify the exact polynomials for each rule.

Typology: Quizzes

Pre 2010

Uploaded on 02/09/2009

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EML3041 – Computational Methods
Sample Quiz Questions
Integration
1. Two segment trapezoidal rule of integration is exact for integrating at most
A. first order polynomials
B. second order polynomials
C. third order polynomials
D. fourth order polynomials
2. Find the value of the integral ∫5
1e-t dt using exact integration?
3. Find the value of the integral ∫5
1e-t dt using 2-segment Trapezoidal rule.
4. Two-point Gauss Quadrature rule is exact for integrating at most
A. first order polynomials
B. second order polynomials
C. third order polynomials
D. fourth order polynomials
5. Find the value of the integral ∫5
1e-t dt using 2-point Gaussian Quadrature rule.
6. The one-point Gauss Quadrature rule is defined as
)()( 11 xfcdxxf
b
a
≈
∫, where bxa
≤
≤
The values of c1 and x1 are found by assuming that the one-point formula is exact for any
first order polynomial. Find c1 in the above one-point Gauss quadrature rule.
7. The one-point Gauss Quadrature rule is defined as
)()( 11 xfcdxxf
b
a
≈
∫, where bxa
≤
≤
The values of c1 and x1 are found by assuming that the one-point formula is exact for any
first order polynomial. Find x1 in the above one-point Gauss quadrature rule.
8. A scientist develops an approximate formula for integration as
pf2

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EML3041 – Computational Methods Sample Quiz Questions Integration

  1. Two segment trapezoidal rule of integration is exact for integrating at most A. first order polynomials B. second order polynomials C. third order polynomials D. fourth order polynomials
  2. Find the value of the integral (^) ∫^51 e-t^ dt using exact integration?
  3. Find the value of the integral (^) ∫^51 e -t^ dt using 2-segment Trapezoidal rule.
  4. Two-point Gauss Quadrature rule is exact for integrating at most A. first order polynomials B. second order polynomials C. third order polynomials D. fourth order polynomials
  5. Find the value of the integral (^) ∫^51 e -t^ dt using 2-point Gaussian Quadrature rule.
  6. The one-point Gauss Quadrature rule is defined as

f ( x ) dx c 1 f ( x 1 )

b a

∫ ≈ , where^ a^ ≤ x ≤ b

The values of c 1 and x 1 are found by assuming that the one-point formula is exact for any first order polynomial. Find c 1 in the above one-point Gauss quadrature rule.

  1. The one-point Gauss Quadrature rule is defined as

f ( x ) dx c 1 f ( x 1 )

b a

∫ ≈ , where^ a^ ≤ x ≤ b

The values of c 1 and x 1 are found by assuming that the one-point formula is exact for any first order polynomial. Find x 1 in the above one-point Gauss quadrature rule.

  1. A scientist develops an approximate formula for integration as

f ( x ) dx c 1 f ( x 1 )

b a

∫ ≈ , where^ a^ ≤ x ≤ b

The values of c 1 and x 1 are found by assuming that the formula is exact for the functions of the form a0 x+a 1 x^2 polynomial. Then the resulting formula would be exact for integrating A. f(x) = 2 B. f(x) = 2 + 3x + 5x^2 C. f(x) = 5x 2 D. f(x) = 2 + 3x

  1. Use two-point Gaussian quadrature rule to integrate (^) ∫

∞ −

− 2

2 e x^ dx. You may make a change of variables before applying the Gaussian quadrature rule.

  1. The following data of the velocity of a body as a function of time is given as follows. Time (s)^0 15 18 22 Velocity(m/s) 22 24 37 25 123 Use Trapezoidal rule with unequal segments OR a better scientific method to find the distance covered by the body from t=15 to 22 seconds.
  2. The following data of the velocity of a body as a function of time is given as follows. Time (s) 0 15 18 22 24 Velocity(m/s) 22 24 37 25 123 Use Trapezoidal rule with unequal segments OR a better scientific method to find the distance covered by the body from t=13 to 18 seconds.
  3. 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 given by

2 0

.

Q 2 π rVdr .

To save money, you are allowed to put only two velocity probes (these probes send the velocity data to the central office in New York, NY via satellite) in the pipe. What radial locations would you suggest for the two velocity probes for the most accurate calculation of the flow rate? 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. Justify your answer.