Trapezoidal Rule

Imagine a truck moving along a highway. To find the distance it travels during a given time interval t, find the area under its velocity–time graph.

Since the speed of the truck is known at discrete time points, this area can not be calculated precisely. Instead, the area is approximated using the trapezoidal rule. To do this, divide the total time interval into n equal subintervals, each with a width of Δt.

In this method, the two endpoints of each subinterval are connected by a straight line, forming trapezoids.

For a given subinterval, which has two vertical sides representing the velocities and a horizontal base representing the time width, the area of this trapezoid is the average of the lengths of its two vertical sides multiplied by the horizontal distance.

Adding the areas of all n trapezoids gives an approximation of the displacement.

After algebraic manipulations and solving this summation, the first and last velocities appear once, while interior velocities appear twice because each belongs to two trapezoids.

After combining like terms, the resulting expression approximates the total displacement over the specified time interval.