Trapezoid Rule in Calculus

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The trapezoid rule is a numerical integration technique used to approximate the definite integral of a function. It’s based on approximating the area under a curve by dividing it into trapezoids and summing up their areas. This method is particularly useful when an exact analytical solution to the integral is difficult or impossible to obtain.

The general form of the trapezoid rule for approximating the integral of a function $f(x)$ over an interval $[a, b]$ is given by:

$\int_{a}^{b} f(x) \, dx \approx \frac{h}{2} [f(a) + 2f(x_1) + 2f(x_2) + \ldots + 2f(x_{n-1}) + f(b)]$

Where:

• $h = \frac{b – a}{n}$ is the width of each subinterval,
• $x_i = a + i \cdot h$ for $i = 0, 1, 2, \ldots, n$ are the endpoints of the subintervals, and
• $n$ is the number of subintervals (or trapezoids) used in the approximation.

The trapezoid rule improves upon the midpoint rule by approximating the function with straight-line segments between adjacent points, making it more accurate for functions that are not well-approximated by straight lines.

To illustrate, let’s consider an example:

Suppose we want to approximate the integral of $f(x) = x^2$ over the interval $[0, 1]$ using the trapezoid rule with $n = 4$ subintervals.

First, we calculate $h = \frac{1 – 0}{4} = 0.25$, and the endpoints of the subintervals are:

• $x_0 = 0$
• $x_1 = 0.25$
• $x_2 = 0.5$
• $x_3 = 0.75$
• $x_4 = 1$

Applying the trapezoid rule formula:

$\int_{0}^{1} x^2 \, dx \approx \frac{0.25}{2} [f(0) + 2f(0.25) + 2f(0.5) + 2f(0.75) + f(1)]$
$= \frac{0.25}{2} [0 + 2(0.25^2) + 2(0.5^2) + 2(0.75^2) + 1^2]$
$= \frac{0.25}{2} [0 + 0.125 + 0.5 + 1.125 + 1]$
$= \frac{0.25}{2} \times 2.75$
$= 0.25 \times 2.75$
$= 0.6875$

So, the trapezoid rule approximation of $\int_{0}^{1} x^2 \, dx$ with 4 subintervals is approximately 0.6875. This value is close to the exact value of $\frac{1}{3}$, demonstrating the effectiveness of the trapezoid rule for numerical integration.

Would you like to explore any specific aspects or applications of the trapezoid rule further?