The Trapezoidal Rule is a way to estimate the area under a curve when we can't easily calculate it using normal math. It's helpful for tricky functions. Here's a simple guide to using the Trapezoidal Rule to find these estimates.

1. Define the Integral

First, pick the function you want to integrate. We call this function $f(x)$. You'll also choose two points, $a$ and $b$, which mark the start and end of the area you want to find.

For example, if you want to find the area under $f(x)$ from $x = a$ to $x = b$, you would write:

$$\int_a^b f(x) \, dx$$

2. Decide on the Number of Divisions (n)

Next, decide how many smaller sections, or subintervals, you want to divide the interval $[a, b]$ into. We call this number $n$.

How many sections you choose will affect how accurate your result is. If you choose more sections (a larger $n$), you get a more accurate answer, but it also means you’ll have to do more calculations.

It's common to choose $n$ to be between 5 to 10 for a good estimate, depending on the function.

3. Calculate the Width of Each Section

Now, you need to find out how wide each section will be. This width is called $h$, and you can find it using this formula:

$$h = \frac{b - a}{n}$$

4. Compute the Function Values at Each Endpoint

Next, you need to figure out the value of your function at the endpoints of each section.

This means you will calculate $f(a)$, $f(a + h)$, $f(a + 2h)$, and so on, until $f(b)$.

Overall, you will perform $n + 1$ calculations at these points: $x_0$, $x_1$, ..., $x_n$, where $x_i = a + ih$ for $i = 0, 1, 2, ..., n$.

5. Use the Trapezoidal Rule Formula

Now it’s time to estimate the area! You can use this formula:

$$\int_a^b f(x) \, dx \approx \frac{h}{2} \left( f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right)$$

In this formula:

• $f(x_0)$ is the function value at the start ($a$).
• $f(x_n)$ is the function value at the end ($b$).
• The middle part ($2 \sum_{i=1}^{n-1} f(x_i)$) adds up the function values for the points in between.

6. Calculate the Approximate Result

Finally, follow the formula from step 5 to calculate an approximate value for the area under the curve. Be sure to add and multiply carefully to get an accurate result.

Example

Let’s say you want to estimate the area under the curve of $f(x) = x^2$ from $0$ to $2$ using $n = 4$.

1. Here, $a = 0$, $b = 2$, and $n = 4$.

2. Calculate $h$:

$$h = \frac{2 - 0}{4} = 0.5$$

3. Compute the function values:

• $f(0) = 0^2 = 0$
• $f(0.5) = (0.5)^2 = 0.25$
• $f(1) = 1^2 = 1$
• $f(1.5) = (1.5)^2 = 2.25$
• $f(2) = 2^2 = 4$

4. Now apply the formula:

$$\int_0^2 x^2 \, dx \approx \frac{0.5}{2} \left( 0 + 2(0.25 + 1 + 2.25) + 4 \right)$$

5. Calculate the result:

$$\approx \frac{0.5}{2} \left( 0 + 2 \times 3.5 + 4 \right) = \frac{0.5}{2} \left( 7 + 4 \right) = \frac{0.5}{2} \times 11 = 2.75$$

So, the estimated area under the curve is about 2.75.