4.1. Basics of Numerical Integration

Try these first

1. 1.

   What is the Trapezoidal Rule used for?

   Hint

   Think about the shape used in the calculation.

2. 2.

   What must be true about the function f(x) for numerical integration?

   Hint

   Consider the behavior of f(x) in the given interval.

3. 3.

   What does numerical integration approximate?

   • The derivative of a function
   • The area under a curve
   • The slope of the function

   Hint

   Focus on the purpose of numerical methods.

4. 4.

   True or False: The Trapezoidal Rule can only be used on linear functions.

   • True
   • False

   Hint

   Think about the flexibility of this rule.

5. 5.

   Given the function f(x) = sin(x) on the interval [0, π] with n = 4, calculate the approximate integral using the Trapezoidal Rule and discuss the expected versus actual error.

   Hint

   Remember, calculate values for the function, and estimate second derivatives.

6. 6.

   Using Simpson's 3/8 Rule, approximate the integral of f(x) = e^x from 0 to 1 with n = 3 intervals, and evaluate the accuracy compared to an exact solution.

   Hint

   Identify function values at required points to compute the approximation.