Simpson's Rule and Numerical Integration

This video tutorial explains numerical integration using Simpson's Rule with n equals four. It covers the setup of Simpson's Rule, including the calculation of delta x and x values. The tutorial demonstrates the application of the Simpson's Rule formula to approximate a definite integral. It also includes a step-by-step guide on evaluating function values using a graphing calculator, leading to the final approximation of the integral.

What is the primary purpose of using Simpson's Rule in numerical integration?

• To approximate definite integrals using linear functions

• To find exact solutions to definite integrals

• To approximate definite integrals using quadratic functions

• To approximate definite integrals using cubic functions

Why must the number of sub-intervals (n) be even when using Simpson's Rule?

• Because it simplifies the calculations

• Because it is a requirement for all numerical methods

• Because quadratic functions span two sub-intervals

• Because it reduces the error in approximation

What is the formula for calculating delta x in Simpson's Rule?

• b - a

• b + a

• (b + a) / n

• (b - a) / n

What is the value of delta x when the limits of integration are 1 and 3, and n equals 4?

• 0.5

• 2.0

• 1.0

• 0.25

What is the pattern of coefficients used in Simpson's Rule for the inner sum?

• 1, 2, 1, 2, 1

• 1, 4, 1, 4, 1

• 2, 4, 2, 4, 2

• 4, 2, 4, 2, 4

Which tool is used to evaluate the function values in this example?

• A computer algebra system

• A spreadsheet software

• A graphing calculator

• A scientific calculator

What is the function used in this example for evaluating the definite integral?

• f(x) = 3√(x^2 + x)

• f(x) = x^2 + x

• f(x) = 3x^2 + x

• f(x) = √(3x^2 + x)

What is the approximate value of f(1) rounded to four decimal places?

• 4.6789

• 4.5678

• 4.1234

• 4.2426

What is the final approximation of the definite integral using Simpson's Rule?

• 14.1234

• 14.5678

• 14.6777

• 14.7890

How many decimal places are preferred for the final result in this example?

• Two decimal places

• Three decimal places

• Five decimal places

• Four decimal places