what is Trapezoidal rule? How does it work in finding the area under the curve(s)?
This probably takes up a whole section in your Chapter on Area & Integration, too large a topic to explain here.
You might want to look at this.
http://www.intmath.com/Integration/5_Trapezoidal-rule.php
The Trapezoidal Rule is a method used to approximate the area under a curve. It is based on dividing the area into a series of trapezoids and adding up their individual areas to get an approximation of the total area.
Here's how the Trapezoidal Rule works:
1. Start with a function for which you want to find the area under its curve, let's call it f(x).
2. Choose the interval over which you want to calculate the area. Let's say the lower limit is 'a' and the upper limit is 'b'.
3. Divide the interval [a, b] into 'n' equal sub-intervals. The more sub-intervals you choose, the more accurate your result will be.
4. Determine the width of each sub-interval, which is given by: Δx = (b - a) / n.
5. Now, for each sub-interval, calculate the height of the trapezoid by evaluating the function at the endpoints of the interval. If the height at the left endpoint is f(a) and the height at the right endpoint is f(b), then the height of the trapezoid is: h = (f(a) + f(b)) / 2.
6. Once you have the width (Δx) and height (h), calculate the area of each trapezoid using the formula: A = Δx * h.
7. Finally, add up the areas of all the trapezoids to get an approximation of the total area under the curve.
The formula for the approximation is: A ≈ (Δx / 2) * (f(a) + 2 * f(x₁) + 2 * f(x₂) + ... + f(b)), where x₁, x₂, ... are the midpoints of each sub-interval.
With more sub-intervals (larger 'n' value), the approximation becomes more accurate, approaching the actual area under the curve.
It's important to note that the Trapezoidal Rule is only an estimation method and may not always provide a perfectly accurate result, especially for functions with complex behaviors. However, it is a simple and relatively efficient method for numerical integration.