#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

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#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 $f(x_{i})$. These are the values of the function at the left-hand and right-hand sides of the trapezoid.
   • The area of the $i$th trapezoid is $(x_{i}-x_{i-1}) \cdot \frac{f(x_{i-1}) + f(x_{i})}{2}$.

3. The trapezoidal sum is the sum of the areas of these $n$ trapezoids: $$\sum_{i=1}^{n} (x_{i}-x_{i-1}) \cdot \frac{f(x_{i-1}) + f(x_{i})}{2}$$

Consider a function $f$ with $n=5$ trapezoids. Increasing the number of trapezoids, $n$, generally gives a more accurate approximation. In exams, you may be given values of the function in a table rather than the function itself.

#Special Case: Equal Intervals

When all intervals in a trapezoidal sum are the same size, the formula simplifies:

• The width of each trapezoid is $\frac{b-a}{n}$.

The trapezoidal sum becomes: $$\sum_{i=1}^{n} \frac{b-a}{n} \cdot \frac{f(x_{i-1}) + f(x_{i})}{2}$$

Simplified further: $$\frac{b-a}{n} \left[ \frac{f(x_{0}) + f(x_{n})}{2} + \sum_{i=1}^{n-1} f(x_{i}) \right]$$ or $$\frac{b-a}{2n} \left[ f(x_{0}) + f(x_{n}) + 2 \sum_{i=1}^{n-1} f(x_{i}) \right]$$

#Overestimation and Underestimation

• If a function is concave up over the interval, the trapezoidal sum will be an overestimate.
• If a function is concave down over the interval, the trapezoidal sum will be an underestimate.

#Worked Example

A social sciences researcher models the total mass of garden gnomes using a function $m$. The table below gives values of $m'(t)$ over the interval $0 \le t \le 12$. At $t=0$, $m(0)=24.9$ kg.

#Calculation:

1. Divide the interval into four subintervals:

   • Widths: $3-0$, $7-3$, $10-7$, $12-10$
   • Heights: $m'(0), m'(3), m'(7), m'(10), m'(12)$

2. Calculate the area of each trapezoid: $$\begin{aligned} &\text{1st Trapezoid:} & (3-0) \cdot \frac{2.6 + 4.8}{2} \\ &\text{2nd Trapezoid:} & (7-3) \cdot \frac{4.8 + 12.2}{2} \\ &\text{3rd Trapezoid:} & (10-7) \cdot \frac{12.2 + 0.7}{2} \\ &\text{4th Trapezoid:} & (12-10) \cdot \frac{0.7 + (-1.3)}{2} \end{aligned}$$

3. Sum the areas: $$\left(3-0\right) \cdot \frac{2.6 + 4.8}{2} + \left(7-3\right) \cdot \frac{4.8 + 12.2}{2} + \left(10-7\right) \cdot \frac{12.2 + 0.7}{2} + \left(12-10\right) \cdot \frac{0.7 + (-1.3)}{2} = 63.85 \text{ kg}$$

4. Add the initial mass: $$63.85 + 24.9 = 88.75 \text{ kg}$$

The total mass at $t=12$ is approximately 88.75 kg.

#Practice Questions

Practice Question

1. Calculate the trapezoidal sum for $f(x)$ from $x=0$ to $x=4$ using 4 subintervals: $f(0)=3$, $f(1)=5$, $f(2)=7$, $f(3)=6$, $f(4)=2$.

2. Determine whether the trapezoidal sum is an overestimate or underestimate for $f(x) = x^2$ over $[0, 2]$.

#Glossary

Trapezoidal Sum: An approximation method for the area under a curve by dividing it into trapezoids.

Definite Integral: The net area under a curve between two points, often represented as $\int_{a}^{b} f(x) , dx$.

Concave Up: A function is concave up if its second derivative is positive.

Concave Down: A function is concave down if its second derivative is negative.

#Summary and Key Takeaways

#Summary

• Trapezoidal sums approximate the area under a curve by summing the areas of trapezoids.
• The formula can be simplified when intervals are of equal size.
• Understanding concavity helps determine if the sum is an overestimate or underestimate.

#Key Takeaways

• Trapezoidal Sum Formula: $$\sum_{i=1}^{n} (x_{i}-x_{i-1}) \cdot \frac{f(x_{i-1}) + f(x_{i})}{2}$$
• Concavity and Estimation:
  • Concave Up: Overestimate
  • Concave Down: Underestimate

#Exam Strategy

1. Understand the Process: Focus on understanding the method rather than memorizing the formula.
2. Check Concavity: Determine if the function is concave up or down to predict overestimation or underestimation.
3. Use Tables Efficiently: Be comfortable using values from tables to calculate trapezoidal sums.

Happy studying!