In general, discrete mathematics is much more relevant in building software than is continuous mathematics.

The purpose of integral calculus is to calculate the length along a line, the area under a curve, the volume in a surface. The area under function f(x) from x₁ to x₂ can be expressed in integral calculus as follows.

• horizontal values: x_i
• vertical values: f(x_i)

Integral calculus is used to calculate the area under a curve.

Discrete approximation: trapezoidal method

Area
= 0.5*(f(x_i)+f(x_i+1))*(x_i+1-x_i)
= 0.5*(f(x_i)+f(x_i+1))*dx

The trapezoidal method is used to approximate the area under the curve by dividing the curve into trapezoids and adding the area of each trapezoid.

The trapezoidal method approximates the area under a curve by approximating the integral with the explicit summation

Consider a sequence of n+1 points (x_i, f(x_i)) where 0 ≤ i ≤ n. Two adjacent points are

(x_i, f(x_i)) and (x_i+1, f(x_i+1)) where 0 ≤ i < n .

The area under the trapezoid formed by these two points is

(x_i+1-x_i) * (f(x_i+1)+f(x_i)) / 2.0 .

So the algorithm to determine the area under the curve is as follows.

Set the area to 0.0
FOR EACH adjacent pair of points (x1,f1) and (x2,f2) DO
add (x2-x1)*(f2+f1)/2.0 to the area
END

In cases where the differences between each of the x_i and x_i+1 is the same, the algorithm may be simplified to the following.

Set the area to 0.0
Set dx to x2-x1
FOR EACH adjacent pair of points (x1,f1) and (x2,f2) DO
add dx*(f2+f1)/2.0 to the area
END

In order to improve the machine performance of the algorithm, the multiplication by (x2-x1) and the division by 2.0 need be done only once.

Set the area to 0.0
FOR EACH adjacent pair of points (x1,f1) and (x2,f2) DO
add f2+f1 to the area
END
Multiply the area by (x2-x1)/2.0

What is the purpose of the trapezoidal method? Give a specific example.