The Trapezoidal Rule
The trapezoidal rule is a numerical method used to approximate the definite integral of a function or the area under a curve.
It's particularly useful when dealing with complex functions or data sets where analytical integration is difficult or impossible.
Basic Concept
- The trapezoidal rule works by dividing the area under a curve into a series of trapezoids.
- The sum of the areas of these trapezoids provides an approximation of the total area under the curve.
The more trapezoids used (i.e., the smaller the width of each trapezoid), the more accurate the approximation becomes.
Formula
For a function $f(x)$ on the interval $[a,b]$, divided into $n$ equal subintervals, the trapezoidal rule is given by:
$$ \int_{a}^{b} f(x) dx \approx \frac{h}{2} [f(x_0) + 2f(x_1) + 2f(x_2) + ... + 2f(x_{n-1}) + f(x_n)] $$
Where:
- $h = \frac{b-a}{n}$ is the width of each subinterval
- $x_0 = a$ and $x_n = b$ are the endpoints of the interval
- $x_1, x_2, ..., x_{n-1}$ are the intermediate points
To apply the trapezoidal rule, consider a simple example: approximating the integral of $f(x)=x^2$ from $a=0$ to $b=4$ using $n=4$ subintervals.
- Calculate the step size:
$$h=\frac{b-a}{n}=\frac{4-0}{4}=1$$ - Determine function values at the points $x_0=0, x_1=1, x_2=2, x_3=3,x_4=4$:
$$f(0)=0^2=0, \quad f(1)=1, \quad f(2)=4, \quad f(3)=9, \quad f(4)=16$$ - Apply the formula:
$$
\begin{aligned}
& \int_0^4 x^2 d x \approx \frac{1}{2}[0+2(1)+2(4)+2(9)+16] \\
& =\frac{1}{2} \times(0+2+8+18+16)=\frac{1}{2} \times 44=22
\end{aligned}
$$ - Thus, the estimated area under $y=x^2$ from 0 to 4 is approximately 22.
Application to Tabular Data
The trapezoidal rule is particularly useful when dealing with discrete data points rather than continuous functions. Given a table of $(x,y)$ values, we can apply the rule as follows:
$$ Area \approx \frac{1}{2} \sum_{i=1}^{n} (x_i - x_{i-1})(y_i + y_{i-1}) $$
Suppose we have the following dataset representing velocity $v(m/s)$ recorded at different times $t(s)$:
| $t(s)$ | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| $v(m/s)$ | 0 | 2 | 5 | 7 | 10 |
Using the trapezoidal rule:
$$
\begin{gathered}
\text { Area } \approx \frac{1}{2} \sum_{i=1}^n\left(t_i-t_{i-1}\right)\left(v_i+v_{i-1}\right) \\
=\frac{1}{2}[(1-0)(2+0)+(2-1)(5+2)+(3-2)(7+5)+(4-3)(10+7)] \\
=\frac{1}{2}[2+7+12+17]=\frac{1}{2} \times 38=19 \mathrm{~m}
\end{gathered}
$$
Thus, the estimated displacement over 4 seconds is 19 meters.
Link to Upper and Lower Bounds
The trapezoidal rule can be related to upper and lower bounds (SL1.6) in the following way:
- If the function is concave up, the trapezoidal approximation provides an upper bound for the true area.
- If the function is concave down, it provides a lower bound.
This relationship can be used to estimate the error in the approximation.
Connection to Areas Under Curves
- The trapezoidal rule is a practical application of the concept of areas under curves (SL5.5).
- While integration provides exact solutions for areas under curves of known functions, the trapezoidal rule allows for approximation of areas under curves that may not have easily integrable functions or are only known through discrete data points.
Applications in Physics and Real-World Scenarios
Kinematics in Physics
In physics, the trapezoidal rule can be used to calculate displacement from a velocity-time graph or work done from a force-displacement graph.
- Given a velocity-time graph for a car's motion, the area under this curve represents the displacement.
- By applying the trapezoidal rule to the data points on this graph, we can approximate the total distance traveled by the car.
Irregular Areas
The trapezoidal rule is particularly useful for calculating areas of irregular shapes, such as lakes or land parcels, which don't conform to simple geometric shapes or mathematical functions.
To estimate the area of a lake:
- Divide the lake into strips of equal width
- Measure the length of each strip at both ends
- Apply the trapezoidal rule to these measurements
This method provides a good approximation of the lake's area without needing to fit it to a mathematical function.
Using Dynamic Graphing Software
Modern graphing calculators and software packages often have built-in functions for applying the trapezoidal rule. These tools allow students to:
- Input functions or data points
- Specify the number of trapezoids or the width of each trapezoid
- Instantly calculate the approximate area
- Visualize the trapezoids overlaid on the curve
Experiment with different numbers of trapezoids to see how the approximation improves with smaller intervals.
Some students mistakenly believe that the trapezoidal rule always overestimates the true area. In reality, it can overestimate or underestimate depending on the shape of the curve.
Another common error is forgetting to halve the sum of the first and last y-values in the formula. Remember, these endpoints are only counted once, while all other points are counted twice.
