Master Numerical Integration Techniques: Quizzes and Practice

Question 1

The number of quadrature points used in a Gaussian quadrature formula is always:

Accepted Answer

Equal to the degree of the polynomial being integrated.

Answer

Even.

Answer

Odd.

Answer

Dependent on the interval of integration.

Question 2

Which of the following quadrature rules offers the highest order of accuracy for numerical integration?

Accepted Answer

Gaussian quadrature

Answer

Trapezoidal rule

Answer

Simpson's rule

Answer

Midpoint rule

Question 3

In which of the following ways is numerical integration commonly used in real-world applications?

Accepted Answer

Approximating areas under curves to calculate quantities such as volume and work

Answer

Solving differential equations directly

Answer

Finding exact solutions to complex integrals

Question 4

Which technique is ineffective in reducing the error in numerical integration?

Accepted Answer

Using a random integration rule

Answer

Adaptive step size control

Answer

Richardson extrapolation

Answer

Increasing the number of subintervals

Question 5

Which of the following is NOT a numerical integration technique used to approximate definite integrals?

Accepted Answer

Euler's method

Answer

Trapezoidal rule

Answer

Simpson's rule

Answer

Gaussian quadrature

Question 6

The Trapezoidal rule approximates the definite integral ∫[a, b] f(x) dx using the formula:

Accepted Answer

(b - a) * [f(a) + f(b)] / 2

Answer

(b - a) * [f(a) - f(b)] / 2

Answer

(b + a) * [f(a) + f(b)] / 2

Answer

(b + a) * [f(a) - f(b)] / 2

Question 7

Simpson's rule achieves higher accuracy than the Trapezoidal rule by utilizing:

Accepted Answer

Higher-degree interpolation polynomials

Answer

Lower-degree interpolation polynomials

Answer

The same degree of interpolation polynomials

Answer

Numerical differentiation

Question 8

Gaussian quadrature employs which mathematical concept to determine integration weights?

Accepted Answer

Orthogonal polynomials

Answer

Trigonometric functions

Answer

Exponential functions

Answer

Random weight generation

Question 9

A notable advantage of Gaussian quadrature is its:

Accepted Answer

Enhanced precision for smooth functions

Answer

Reduced computational cost compared to Simpson's rule

Answer

Simplicity of implementation

Answer

Ability to integrate functions with singularities

Question 10

When assessing the accuracy of numerical integration, which method is commonly employed?

Accepted Answer

Richardson extrapolation

Answer

Monte Carlo simulation

Answer

Quadrature by parts

Answer

Integral transforms

Question 11

In the context of numerical integration, the step size refers to:

Accepted Answer

The width of each subinterval

Answer

The total number of subintervals

Answer

The value of the integrand at the midpoint of each subinterval

Answer

The allowable error tolerance in the approximation

Question 12

What is a primary benefit of adaptive numerical integration methods over fixed-order methods?

Accepted Answer

Enhanced accuracy with equivalent computational effort

Answer

Lower computational cost than fixed-order methods

Answer

Simpler implementation than fixed-order methods

Question 13

Which of the following is the formula for the Trapezoidal Rule for approximating the definite integral ∫ab​f(x)dx?

Accepted Answer

∫ab​f(x)dx≈h/2[f(a)+f(b)]

Answer

∫ab​f(x)dx≈h/6[f(a)+4f(c)+f(b)]

Answer

∫ab​f(x)dx≈h/12[f(a)+2f(m)+f(b)]

Answer

∫ab​f(x)dx≈h/3[f(a)+3f(m)+f(b)]

Question 14

What is the order of convergence of the Trapezoidal Rule?

Accepted Answer

O(h^2)

Answer

O(h)

Answer

O(h^3)

Answer

O(log h)

Question 15

Which of the following is a disadvantage of the Trapezoidal Rule?

Accepted Answer

It is not suitable for highly oscillatory functions.

Answer

It requires function evaluations only at the endpoints.

Answer

It always overestimates the integral.

Answer

It has a high order of convergence.

Question 16

What is the order of convergence of Simpson's Rule?

Accepted Answer

O(h^4)

Answer

O(h^2)

Answer

O(h^3)

Answer

O(h^5)

Question 17

Which of the following is the formula for the Gaussian quadrature rule for approximating the definite integral ∫ab​f(x)dx?

Accepted Answer

∫ab​f(x)dx≈h∑i=1n​wi​f(xi​)

Answer

∫ab​f(x)dx≈h/6[f(a)+4f(b)+f(c)]

Answer

∫ab​f(x)dx≈h/12[f(a)+2f(m)+f(b)]

Answer

∫ab​f(x)dx≈h/3[f(a)+3f(m)+f(b)]

Question 18

How are the weights in the Gaussian quadrature rule determined?

Accepted Answer

Computed from the zeros of a polynomial.

Answer

Equal to the coefficients of the basis functions.

Answer

Always positive.

Answer

Dependent on the order of the rule.

Question 19

Which formula accurately represents the Trapezoidal rule for estimating a definite integral?

Accepted Answer

(b - a) / 2 * (f(a) + f(b))

Answer

(b - a) / 2 * (f(a) - f(b))

Answer

(b - a) * (f(a) + f(b))

Answer

(b - a) * (f(a) - f(b))

Question 20

What is the order of accuracy achieved by Simpson's rule?

Accepted Answer

3

Answer

1

Answer

2

Answer

4

Question 21

Which advantage sets Gaussian quadrature apart from other numerical integration techniques?

Accepted Answer

Enhanced accuracy for smooth functions

Answer

Simplified implementation

Answer

Faster convergence

Answer

Reduced computational cost

Question 22

In science and engineering, what are some key applications of numerical integration?

Accepted Answer

Approximating fluid flow patterns

Accepted Answer

Solving differential equations numerically

Accepted Answer

Calculating areas of irregularly shaped objects

Answer

Finding roots of polynomial equations

Question 23

The Trapezoidal rule approximates the area under a curve by:

Accepted Answer

Dividing the curve into trapezoids and summing their areas

Answer

Fitting a linear function to the curve and integrating it

Answer

Using a weighted average of the function values at the endpoints of the curve

Answer

Using a Gaussian distribution to model the curve and integrating it

Question 24

Gaussian quadrature uses:

Accepted Answer

A set of pre-determined weights and nodes to approximate the integral

Answer

A weighted average of the function values at the endpoints of the interval

Answer

A higher-order polynomial to approximate the function on each subinterval

Answer

A Gaussian distribution to model the function and integrate it

Question 25

Which of the following functions is best suited for integration using the Trapezoidal rule?

Accepted Answer

A function with a smooth curve

Answer

A function with a sharp peak

Answer

A function with a discontinuity

Answer

A function with a rapidly oscillating derivative

Question 26

Gaussian quadrature is often used to approximate integrals of functions that are:

Accepted Answer

Smooth and well-behaved

Answer

Have sharp peaks or discontinuities

Answer

Oscillate rapidly

Answer

Have a rapidly oscillating derivative

Question 27

A potential drawback of using Gaussian quadrature is its:

Accepted Answer

Computational cost

Answer

Limited applicability to specific functions

Answer

Lower accuracy compared to other numerical integration techniques

Answer

Difficulty in implementation

Question 28

Which numerical integration method possesses the highest order of accuracy?

Accepted Answer

Gaussian Quadrature

Answer

Trapezoidal Rule

Answer

Simpson's Rule

Answer

Monte Carlo Integration

Question 29

Which of the following is NOT a commonly used numerical integration technique?

Accepted Answer

Monte Carlo integration

Answer

Trapezoidal rule

Answer

Gaussian quadrature

Answer

Simpson's rule

Question 30

Which of the following statements best describes the Trapezoidal rule?

Accepted Answer

It approximates the definite integral by dividing the interval into subintervals of equal width and using trapezoids to estimate the area under the curve.

Answer

It uses a set of weighted points to approximate the definite integral of a function.

Question 31

What degree polynomial does Simpson's rule use in its approximation?

Accepted Answer

3

Answer

2

Answer

4

Answer

1

Question 32

Which factor is NOT used to estimate the error in numerical integration?

Accepted Answer

The number of evaluations

Answer

The order of the numerical integration method

Answer

The smoothness of the function being integrated

Answer

The width of the integration interval

Question 33

Which of the following is the error formula for the Trapezoidal rule?

Accepted Answer

-(h^2/12) * f''(xi)

Answer

-(h^3/2) * f'''(xi)

Answer

-(h^3/12) * f'''(xi)

Answer

-(h^2/2) * f''(xi)

Question 34

Which numerical integration technique is known for its exceptionally high accuracy?

Accepted Answer

Gaussian quadrature

Answer

Trapezoidal rule

Answer

Monte Carlo integration

Answer

Simpson's rule

Question 35

Which of the following is NOT an advantage of numerical integration?

Accepted Answer

Exact results for all functions

Answer

Approximations of definite integrals

Answer

Flexibility in handling complex functions

Answer

Error analysis and control

Question 36

The Trapezoidal rule approximates the integral by dividing the area under the curve into:

Accepted Answer

Trapezoids

Answer

Parallelograms

Answer

Rectangles

Answer

Triangles

Question 37

The Composite Trapezoidal rule is obtained by:

Accepted Answer

Applying the Trapezoidal rule to subintervals

Answer

Integrating the Trapezoidal rule over the entire interval

Answer

Using a higher-order polynomial approximation

Answer

Halving the step size

Question 38

The weights in Gaussian quadrature are:

Accepted Answer

Dependent on the chosen nodes

Answer

Equal to the corresponding function values

Answer

Constant for all functions

Question 39

Which numerical integration technique is generally considered the most accurate?

Accepted Answer

Gaussian quadrature

Answer

Trapezoidal rule

Answer

Simpson's rule

Answer

Midpoint rule

Question 40

What is the order of convergence of the Trapezoidal rule for approximating definite integrals?

Accepted Answer

O(h^2)

Answer

O(h)

Answer

O(h^3)

Answer

O(h^4)

Question 41

Which of the following is NOT an advantage of using numerical integration?

Accepted Answer

Increased accuracy

Answer

Time savings

Answer

Convenience for problems without closed-form solutions

Answer

Flexibility in handling complex functions

Question 42

Considering both computational cost and accuracy, which numerical integration technique is typically the most efficient?

Accepted Answer

Gaussian quadrature

Answer

Midpoint rule

Answer

Trapezoidal rule

Answer

Simpson's rule

Question 43

What is the fundamental distinction between numerical integration and analytical integration?

Accepted Answer

Numerical integration approximates definite integrals using numerical methods, while analytical integration finds exact solutions.

Answer

Analytical integration can only be applied to simple functions.

Answer

Numerical integration is only used when closed-form solutions are not available.

Answer

Numerical integration is more accurate than analytical integration.

Question 44

Which of the following is NOT a recognized numerical integration technique?

Accepted Answer

Derivative calculus

Answer

Trapezoidal rule

Answer

Gaussian quadrature

Answer

Simpson's rule

Question 45

Which formula represents the composite Simpson's rule?

Accepted Answer

∫[a,b] f(x) dx ≈ h/3 [f(a) + 4f(a+h) + 2f(a+2h) + ... + 4f(b-h) + f(b)]

Answer

∫[a,b] f(x) dx ≈ h/6 [f(a) + 4f(a+h) + 2f(a+2h) + ... + 2f(b-h) + 4f(b)]

Answer

∫[a,b] f(x) dx ≈ h [f(a) + f(a+h) + f(a+2h) + ... + f(b-h) + f(b)]

Answer

∫[a,b] f(x) dx ≈ h/2 [f(a) + 2f(a+h) + 2f(a+2h) + ... + 2f(b-h) + f(b)]

Question 46

Gaussian quadrature utilizes what principle for integration?

Accepted Answer

Weighted sum of function values at specific points

Answer

Trapezoidal rule

Answer

None of the above

Answer

Midpoint rule

Question 47

Which of the following is a key advantage of Gaussian quadrature compared to other numerical integration techniques?

Accepted Answer

Higher accuracy for a given number of function evaluations

Answer

Faster computation time

Answer

Easier to implement

Question 48

Error analysis in numerical integration involves estimating the:

Accepted Answer

Difference between the numerical approximation and the true value

Answer

Computational time

Answer

Number of subintervals needed

Question 49

Which of the following is NOT a factor that affects the accuracy of numerical integration?

Accepted Answer

The choice of integration formula

Answer

The smoothness of the function

Answer

The size of the interval

Answer

The number of subintervals

Question 50

Numerical integration can be used to:

Accepted Answer

Approximate areas under curves

Answer

Invert matrices

Answer

Find roots of polynomials

Answer

Solve differential equations