Trapezoidal rule is a technique used to approximate the value of a definite integral ($\int_{a}^{b}$ f(x)dx). Supposing f(x) is continuous on [a, b] divide [a, b] into n sub intervals of equal length $\bigtriangleup$x = $\frac{b - a}{n}$.  Using n + 1 points we get $x_{0}$ = a, $x_{1}$ = a + $\bigtriangleup$x, $x_{2}$ = a + 2$\bigtriangleup$x, ......, $x_{n}$ = a + n$\bigtriangleup$x = b.

f(x) at these points is $y_{0}$ = f($x_{0}$), $y_{1}$ = f($x_{1}$),.......,$y_{n}$ = f($x_{n}$). As straight lines are formed between the points (x$_{i-1}$, y$_{i-1}$) and (x$_{i}$, y$_{i}$) for 1 $\leq$ i $\leq$ n approximate the integral using n trapezoids.

Adding the area of a rectangle and a triangle the area of trapezoid is

A = $y_{0}$$\bigtriangleup$x + $\frac{1}{2}$ ($y_{1} - y_{0}$) $\bigtriangleup$x

A = $\frac{(y_{0} + y_{1})\bigtriangleup x}{2}$

## Trapezoidal Rule Formula

Area of trapezoid is A = $\frac{(y_{0} + y_{1})\bigtriangleup x}{2}$
Adding the area of n trapezoids the approximation is

$\int_{a}^{b}f(x)dx$  $\approx$  $\frac{(y_{0}+y_{1})\bigtriangleup x}{2}$ + $\frac{(y_{1}+y_{2})\bigtriangleup x}{2}$ + ........ + $\frac{(y_{n-1} + y_{n})\bigtriangleup x}{2}$

which is simplified to

$\int_{a}^{b}f(x)dx$  $\approx$ $\frac{\bigtriangleup x}{2}$ ($y_{0} + 2y_{1}+ 2y_{2}+ ...... + 2y_{n-1}+y_{n})$
Therefore the formula for trapezoidal rule is
$\int_{a}^{b}f(x)dx$  $\approx$ $\frac{\bigtriangleup x}{2}$ ($y_{0} + 2y_{1}+ 2y_{2}+ ...... + 2y_{n-1}+y_{n})$

### Integral Table

 Numerical Integration Trapezoidal Rule A Trapezoid Trapezoid Prism Integrability Definite Integral Derivative of an Integral Double Integrals Examples of Integration How to do Indefinite Integrals How to do Integration by Parts Improper Integral Integral by Substitution
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