Using the Trapezoidal Rule in Calculus Made Simple

The Trapezoidal Rule helps us find an estimate of the area under a curve. Here’s how to use it step by step.

What is the Trapezoidal Rule?
The Trapezoidal Rule approximates the area under a curve using trapezoids instead of rectangles.

The formula looks like this:

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

In this formula:

• $n$ is how many sections (or subdivisions) you decide to use.
• $x_i$ stands for the points between $a$ and $b$.

Step 1: Set the Limits and Subdivisions
First, find your starting point ($a$) and your ending point ($b$).

Next, choose how many subdivisions ($n$) you want.

The more subdivisions you use, the more accurate your answer will be.

But remember, using too many can make calculations harder.

A good starting point is to use $n = 4$ or $n = 8$. You can change it later if you need to be more precise.

Step 2: Calculate Function Values
Now it’s time to find the function values at each point.

1. Calculate $f(a)$ (the value at the starting point) and $f(b)$ (the value at the ending point).
2. For each section, calculate $f(x_i)$, using this formula:
   • $x_i = a + i \cdot \Delta x$
   • Here, $\Delta x = \frac{b-a}{n}$ tells you how wide each section is.

Step 3: Use the Trapezoidal Rule Formula
Plug all the values you found into the Trapezoidal Rule formula.

Make sure to do your math carefully to avoid mistakes.

Step 4: Check Your Error
It's also important to see how accurate your estimate is.

You can evaluate the error like this:

$$\left| E_T \right| \leq \frac{(b-a)^3}{12n^2} M$$

In this, $M$ represents the biggest value of the second derivative of $f$ in the range from $a$ to $b$.

If you follow these steps closely, you’ll be able to use the Trapezoidal Rule to effectively estimate areas under curves in your calculus problems!