## Overview of Trapezoidal Rule Calculator:

A simple calculator made for students, mathematicians, teachers
and researchers. This **Trapezoidal Rule calculator** takes the
function as input and gives the solution. The Trapezoidal Rule
Calculator is all-in-one calculator it not only calculates area
but also the error.

## What is Trapezoidal Rule?:

Before integration, mathematicians used to
wonder how to calculate the area under the parabola. A parabola is
a curve on a plane. Since it is easy to calculate the area of a
rectangle, mathematicians would divide the curve into different
rectangular segments. But soon they realised that it was not a
proper way to do it. There is some area which is not considered or
should not be considered. As you can see from the below graph,
when drawing a rectangle, the top corners are either outside or
inside. Hence they could not calculate more accurately.

By drawing trapezoid instead of rectangle,
we can get more accurate result. The Trapezoidal Rule is based on
Newton-Cotes Formula which is as follows:

where

The results can be improved by partitioning
the integration interval and using the trapezoidal rule to all
subintervals and summing up the results.

### Example of How-to Use The Trapezoidal Rule
Calculator:

Consider the function

calculate the area under the curve for

*n*=8. Enter the function and limits on the calculator
and below is what happens in the background.

**Solution:** Given that

*n*=8 we have

Hence we
will be plotting intervals are 0.5 gaps. Now we will calculate area
for each trapezoid, i.e, for x

_{0}, x

_{1} ,x

_{2},
x

_{3}, x

_{4}, x

_{5}, x

_{6}, x

_{7}.
x

_{0}=1.

Now Applying the Trapezoid Rule formula:

## Alternative methods to Trapezoidal Rule:

There are many other methods alternative to Trapezoidal Rule and
share the common family of numeral integrals. The Simpson’s Rule
is another effective method and has faster convergence than the
former for continuously differentiable functions, though not in
all cases.

Trapezoidal is nearly accurate when used on
periodic functions which are integrated over periodic intervals.
For non-periodic functions, like the Gaussian quadrature and
Clenshaw-Curtis quadrate with an unequally spaced point are more
accurate.

## Types of Integration Approximation calculator

- Midpoint Rule calculator
- Trapezoidal Rule calculator
- Simpson’s Rule calculator
- Left Rectangular Approximation Calculator
- Right Rectangular Approximation Calculator
- Approximation Calculator