6.10
Trapezoidal Rule
Estimating the distance traveled by a vehicle using its recorded velocity over time is a common problem in physics and engineering. When velocity data…
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.
View the full transcript and gain access to JoVE Core videos
Q1: How does the trapezoidal rule approximate the area under a curve?
The trapezoidal rule divides the total interval into equal subintervals and connects the endpoints of each subinterval with a straight line, forming trapezoids. The area of each trapezoid is calculated as the average of its two vertical sides multiplied by the horizontal width. Summing all trapezoid areas yields an approximation of the total area under the curve, making it a key method in approximate integration.
Q2: Why do interior velocities appear twice in the trapezoidal rule formula?
Interior velocities appear twice because each interior point belongs to two adjacent trapezoids. The first and last velocities appear only once, at the boundaries of the total interval. This pattern emerges when combining like terms after summing all individual trapezoid areas.
Q3: What is the practical application of the trapezoidal rule in motion problems?
The trapezoidal rule estimates displacement when velocity data is available at discrete time intervals rather than as a continuous function. By approximating the area under a velocity-time graph, it calculates the total distance traveled by a vehicle or object over a specified time period, making it valuable for analyzing empirical motion data.
Q4: How does dividing the interval into more subintervals affect the trapezoidal rule approximation?
Increasing the number of equal subintervals improves the accuracy of the trapezoidal rule approximation. With more subintervals, the straight lines connecting endpoints more closely follow the actual curve, resulting in a more reliable estimate of the total area or displacement, especially for time-dependent processes.
Q5: What does each component of a trapezoid represent in a velocity-time graph?
In a velocity-time graph, each trapezoid has two vertical sides representing the velocities at the beginning and end of a time segment, and a horizontal base representing the time interval width. The area of this trapezoid represents the approximate displacement during that specific time segment.
Q6: When is the trapezoidal rule preferred over other numerical integration methods?
The trapezoidal rule is a fundamental technique in numerical analysis preferred when empirical data is available at discrete points and a continuous function is unavailable. It provides a practical, straightforward approach to integrating motion or other time-dependent processes without requiring complex calculations or assumptions about underlying function behavior.