Calculus II:

Definite integral
numerical approaches
Trapezoidal rule

Trapezoidal rule is another numerical approach to evaluate a definite integral.

In this method, we use trapezoids (or trapeziums). It gives a better approximation to the area under the curve than the rectangle method.

Trapezoidal rule

The area of a the first trapezoid is:

A1 = (y0 + y1)Δx/2

Δx is the « small change in x ».

The area of a the second trapezoid is:

A1 = (y1 + y2)Δx/2

An so on ...

So the approximate area under the curve is found by adding the area of the trapezoids.

A = A1 + A1 + A2 + A3 + A4 + A5 =
(y0 + 2y1 + 2y2 + 2y3 + 2y4 + y5 )Δx/2

For n trapezoids we will have:

A = (y0 + 2y1 + 2y2 + 2y3 + ... + 2y_n-1 + yn )Δx/2

A = Δx (y0/2 + y1 + y2 + y3 + ... + yn/2)

With

Δx = (b - a)/n
y0 = f(a)
yn = f(b)
yk = y0 + kΔx


Note that we get a better approximation if we take more trapezoids . In this Δx will tend to 0.

Finally, if the curve of the function f is above the x-axis between x = a and x = b

From a to b : Area = ∫ f(x) dx = Δx (y0/2 + y1 + y2 + y3 + ... + yn/2)