## Trapezoidal Rule

### The Trapezoidal Rule is a numerical integration technique used to approximate the definite integral of a function

The Trapezoidal Rule is a numerical integration technique used to approximate the definite integral of a function. It is based on dividing the area under the function curve into trapezoids and summing up their areas.

To understand how the Trapezoidal Rule works, let’s consider the integral of a function f(x) over an interval [a, b]. We’ll start by dividing the interval into n subintervals of equal width. The width of each subinterval, Δx, is given by Δx = (b – a) / n.

The Trapezoidal Rule estimates the definite integral by approximating the function curve within each subinterval with a straight line segment. The area of each trapezoid formed is then calculated by taking the average of the heights of the two endpoints and multiplying it by the base (Δx).

The formula for estimating the definite integral using the Trapezoidal Rule is:

∫[a, b] f(x) dx ≈ (Δx / 2) * [f(a) + 2f(x1) + 2f(x2) + … + 2f(xn-1) + f(b)]

where f(x1), f(x2), …, f(xn-1) are the function values at the interior points of the subintervals.

To illustrate the calculations, let’s use an example. Suppose we want to estimate the integral of the function f(x) = x^2 over the interval [0, 2] using the Trapezoidal Rule with n = 4 subintervals.

Step 1: Calculate Δx

Δx = (b – a) / n

Δx = (2 – 0) / 4

Δx = 0.5

Step 2: Evaluate the function at the interval endpoints

f(a) = f(0) = 0^2 = 0

f(b) = f(2) = 2^2 = 4

Step 3: Evaluate the function at the interior points

f(x1) = f(0.5) = (0.5)^2 = 0.25

f(x2) = f(1) = (1)^2 = 1.0

f(x3) = f(1.5) = (1.5)^2 = 2.25

Step 4: Apply the Trapezoidal Rule formula

∫[0, 2] x^2 dx ≈ (Δx /2) * [f(a) + 2f(x1) + 2f(x2) + 2f(x3) + f(b)]

≈ (0.5 / 2) * [0 + 2(0.25) + 2(1.0) + 2(2.25) + 4]

≈ (0.25) * [0 + 0.5 + 2 + 4.5 + 4]

≈ (0.25) * 11

≈ 2.75

Therefore, the estimate of the definite integral of f(x) = x^2 over [0, 2] using the Trapezoidal Rule with n = 4 subintervals is approximately 2.75.

It’s worth mentioning that the accuracy of the Trapezoidal Rule increases as the number of subintervals (n) increases. So, if we were to use more subintervals, we would generally get a more precise estimation. However, it’s important to keep in mind that the Trapezoidal Rule is still an approximation method and cannot provide an exact solution.

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