The trapezoidal rule is another method for approximating the area under curves. It provides a more accurate estimate compared to the rectangle approximation method.

To use the trapezoidal rule, we divide the area under the curve into a series of trapezoids. Each trapezoid is formed by connecting two adjacent points on the curve with straight lines. The formula for calculating the area of a trapezoid is given by A = (b1 + b2) * h / 2, where b1 and b2 are the lengths of the parallel sides and h is the height.

Let's look at an example. Consider the curve y = x^2 + 1 between x = 0 and x = 2. To approximate the area under this curve using the trapezoidal rule, we first divide the interval [0, 2] into n equal sub-intervals. The width of each sub-interval is given by h = (2 - 0) / n.

Next, we calculate the areas of the trapezoids formed by connecting consecutive points on the curve. Finally, we sum up these areas to get an approximation of the total area under the curve. The more sub-intervals we use (i.e., the smaller the value of h), the more accurate our approximation becomes.

Remember, the trapezoidal rule provides a good estimate of the area under curves, but it's still an approximation. For highly curved or irregularly shaped curves, other numerical integration methods may be more suitable. Keep practicing and exploring different techniques to become a master of finding areas under curves!