MAT S213 Reviewer 2

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

• To approximate the area under a curve

• To find the exact value of an integral

• To solve differential equations

• To calculate the derivative of a function

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

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

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

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

• The number of subintervals used in the approximation method

• 3.5

• 3.0

• 2.5

• 2.75

When using Trapezoidal rule, the number of subintervals must be:

• A prime number

• An even integer

• An odd integer

• Any positive integer

Which of the following is true about Riemann sums?

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

• They can only be used with continuous functions.

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

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

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

• The function becomes more quadratic

• The function becomes more linear

• The width of each subinterval increases

• The number of subintervals increases

Which of the following statements is true regarding the use of numerical integration methods?

• Numerical integration methods can only approximate integrals of polynomial functions.

• Numerical integration methods are always less accurate than finding the exact integral.

• Numerical integration methods are useful for approximating integrals when the antiderivative of the function is difficult or impossible to find.

• Numerical integration methods are unnecessary when the antiderivative of the function can be easily found.

The Trapezoidal Rule and Simpson's Rule are examples of:

• Differential calculus techniques

• Numerical integration methods

• Algebraic methods for solving equations

• Analytical geometry tools

When calculating the numerical approximation of an integral using the Trapezoidal Rule, which of the following factors does not affect the accuracy of the approximation?

• The smoothness of the function being integrated

• The method used to calculate the function's values at the endpoints of the subintervals

• The number of subintervals

• The color of the graph of the function

What is the area of a rectangle with length a and width b?

• a²

• a + b

• ab

• b²