4.2. Trapezoidal Rule

Try these first

1. 1.

   What is the Trapezoidal Rule used for?

   Hint

   Think about calculating areas under curves where direct integration is hard.

2. 2.

   What does 'h' stand for in the Trapezoidal Rule formula?

   Hint

   It's derived from the total width divided by the number of intervals.

3. 3.

   What does the Trapezoidal Rule primarily estimate?

   • Area under the curve
   • Rate of change
   • Average value

   Hint

   Consider what we are finding when we integrate.

4. 4.

   True or False: The Trapezoidal Rule gives exact results for all continuous functions.

   • True
   • False

   Hint

   Remember the method relies on approximating regions.

5. 5.

   Calculate the approximate integral of f(x) = ln(x) from 1 to 3 using the Trapezoidal Rule with n = 4. Show calculations.

   Hint

   Remember to evaluate the function at the partition points and apply the trapezoidal formula.

6. 6.

   Consider the function f(x) = cos(x) over the interval [0, π/2]. Apply the Trapezoidal Rule with n = 6 and analyze the convergence of the results as n increases.

   Hint

   Monitor how your approximation improves with more sections!