Numerical Integration Quiz

What is the primary purpose of using Riemann sums in numerical integration?

• To find the exact value of an integral

• To approximate the area under a curve

• To calculate the derivative of a function

• To solve differential equations

Answer explanation

Riemann sums are used in numerical integration to approximate the area under a curve, not to find the exact value of an integral or calculate derivatives.

Answer explanation

The correct formula for the Trapezoidal Rule is the second option, which involves averaging the function values at the endpoints and doubling the sum of the function values at the interior points.

Simpson's Rule is a numerical method that approximates the value of a definite integral by using:

• Linear functions

• Quadratic functions

• Cubic functions

• Quartic functions

Answer explanation

Simpson's Rule uses quadratic functions to approximate the value of a definite integral, making the correct choice Quadratic functions.

Which of the following best describes the error estimation in numerical integration?

• The difference between the exact value of the integral and its numerical approximation

• The sum of the absolute values of the derivatives of the function being integrated

• The maximum value of the function over the interval of integration

• The number of subintervals used in the approximation method

Answer explanation

The error estimation in numerical integration is the difference between the exact value of the integral and its numerical approximation, making this the best description among the options provided.

• 2.75

• 2.5

• 2.0

• 3.0

Answer explanation

The Trapezoidal Rule with 4 subintervals gives an approximation of 2.5 for the definite integral of x^2 from 0 to 2.

When using Simpson's Rule, the number of subintervals must be:

• Any positive integer

• An odd integer

• An even integer

• A prime number

Answer explanation

Simpson's Rule requires the number of subintervals to be an even integer for accurate approximation of integrals.

Which of the following is true about Riemann sums?

• They provide an exact value for the area under a curve.

• They can only be used with continuous functions.

• They can approximate the area under a curve by summing the areas of rectangles.

• They are more accurate than the Trapezoidal Rule and Simpson's Rule for all functions.

Answer explanation

Riemann sums approximate the area under a curve by summing the areas of rectangles, making this statement true.

For approximating definite integrals, the error in the Trapezoidal Rule generally decreases as:

• The function becomes more linear

• The number of subintervals increases

• The width of each subinterval increases

• The function becomes more quadratic

Answer explanation

The error in the Trapezoidal Rule generally decreases as the number of subintervals increases, making it the correct choice.

• 2

• 1.5

• 2.094

• 2.5

Answer explanation

Simpson's Rule with 6 subintervals gives an exact result for integrating sin(x) from 0 to pi, resulting in an approximation of 2.

The error estimation for Simpson's Rule is proportional to:

• The fourth derivative of the function being integrated

• The square of the number of subintervals

• The cube of the width of each subinterval

• The fifth derivative of the function being integrated

Answer explanation

The error estimation for Simpson's Rule is proportional to the fourth derivative of the function being integrated, making it the correct choice.