#Trapezoidal Sums Study Notes

#Table of Contents

1. Introduction to Trapezoidal Sums
2. Calculating a Trapezoidal Sum
3. Special Case: Equal Intervals
4. Overestimation and Underestimation
5. Worked Example
6. Practice Questions
7. Glossary
8. Summary and Key Takeaways

#Introduction to Trapezoidal Sums

#What is a Trapezoidal Sum?

A trapezoidal sum is a method for approximating the exact value of an accumulation of change.

• It is used to approximate the exact value of a definite integral or the exact area between a curve and the x-axis.
• The approximation is achieved by adding up the areas of a number of trapezoids.

#Calculating a Trapezoidal Sum

To calculate the trapezoidal sum of a function $f$ between $x=a$ and $x=b$ (where $a<b$):

1. Divide the interval into $n$ subintervals by choosing values $x_{0}, x_{1}, \ldots, x_{n}$ such that $a=x_{0}<x_{1}<\cdots <x_{n}=b$.

   • The intervals do not need to be the same size.

2. Define $n$ trapezoids:

   • The $i$th trapezoid has a width of $x_{i}-x_{i-1}$. This is the distance from the left-hand side to the right-hand side of the trapezoid.
   • The parallel sides of the $i$th trapezoid have heights of $f(x_{i-1})$ and ...