# Trapezoidal Method

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The **Trapezoidal Method**, also known as the **Trapezoidal Rule**, is an approximation method of numerical integration, and is a member of the closed type group of the Newton-Cotes formulae.

## BackgroundEdit

The Trapezoidal Method is used to approximate the values of *definite integrals*, defined as the area under the graph of the function with respect to (meaning if is negative, then has a negative area), over the compact interval where < . Examples of definite integrals are and .

The Trapezoidal Method belongs to a group of numerical integration formulae called the *Newton-Cotes formulae*, named after Isaac Newton and Roger Cotes. Other formulae belonging to the group (for the *closed* type, of which the Trapezoidal Method is one) include the *Simpson's 1/3* and *3/8 Rules*, and the *Boole's Rule*.

There are two types of applications of the Trapezoidal Method. One is for evalutaion at only two points, and the other is for evalutaion at multiple points -- which is called the **Composite Trapezoidal Method**, For this particular discussion, we will focus on the latter.

## ApplicationsEdit

There are arguably countless applications for integration throughout the many fields of engineering. Among the more common examples of which include finding the velocity of a body from an acceleration function, and finding the displacement of a body from a velocity function. The use of the Trapezoidal Method and other approximation methods for integration could prove to be an effective tool in such applications.

## MethodEdit

Consider the definite integral

- .

We assume that is continuous on into subintervals of equal length, defined as

- ,

and using the points

- ,

after which we can compute for f(x) at these points.

We then form number of trapezoids by drawing straight line segments in between the points and for , as shown below.

Recalling the formula in solving for the area of a trapezoid,

- .

To solve for the area between two subintervals under the curve, defined as , we replace and with and for , and replace with , thus

- .

To solve for the entire area under the curve, defined as , we simply get the sum of all the areas between the subintervals.

Expanding the equation,

and further simplifying it, we get

- .

## ExampleEdit

Use the trapezoidal rule with to estimate

- .

Compute also for percentage of error.

*Solution*

For , we have

We compute for the values of .

Therefore,

To solve for the percentage of error, we compute for the exact value of the integral.

Percentage of error can then be computed using the formula

- .

Substituting the values, we get:

## ImplementationsEdit

### C++Edit

double trapezoidal(double l, double u, int n) { double interval, timestwo, sum; double x[n+1], y[n+1]; int ctr1, ctr2, ctr3; //computes for the interval interval = (u - l) / n; //sets the values for x and f(x) for(ctr1 = 0; ctr1 <= n; ctr1++) { x[ctr1] = l; y[ctr1] = function(l); l = l + interval; } //displays the x and f(x) values cout << "\nx\tf(x)\n----------------\n"; for(ctr2 = 0; ctr2 <= n; ctr2++) cout << x[ctr2] << "\t" << y[ctr2] << "\n"; //adds the values in between the lower and upper limits for(ctr3 = 1; ctr3 < n; ctr3++) { sum = sum + y[ctr3]; } //computes for the entire approximation sum = ((sum * 2) + y[0] + y[n]) * (interval / 2); return sum; }

### JavaEdit

private static double Integrate(double sizeofPanels, double yTable[]){ double firstY = yTable[0]; double lastY = yTable[yTable.length-1]; System.out.println(""); double middle = 0.0; double middleValues = 0.0; for(int i=1; i<yTable.length-1; i++) middleValues = middleValues + yTable[i]; middle = 2*middleValues; double integratedF = (sizeofPanels/2)*(firstY + middle + lastY); return integratedF; }

## ReferencesEdit

Trapezoidal rule - Wikipedia, the free encyclopedia

Trapezoidal Rule: Integration - Holistic Numerical Methods Institute