# Trapezoidal Rule

Top

## Trapezoidal Rule Examples

### Solved Examples

Question 1: Use the trapezoidal rule with n = 5 estimate  $\int_{1}^{5}\sqrt{1+x^{2}}$ dx
Solution:

For n = 5 we have $\bigtriangleup$x = $\frac{5-1}{5}$ = 0.8

Computing the values for $y_{0}$, $y_{1}$,......,$y_{5}$

 x y = $\sqrt{1+x^{2}}$ 1 1.41 1.8 2.06 2.6 2.78 3.4 3.54 4.2 4.32 5 5.10
$\int_{1}^{5}\sqrt{1+x^{2}}$ dx  = $\frac{0.8}{2}$ (1.41 + 2(2.06) + 2(2.18) + 2(3.54) + 2(4.32) + 2(5.10))
= 14.324

Question 2: Use the trapezoidal rule with n = 4 estimate $\int_{1}^{6}x^{4}$ dx.
Solution:

Given a = 1, b = 6 and n = 4  so h = $\frac{b-a}{n}$ = $\frac{6-1}{4}$ = 1.25

$x_{0}$ = 1, $x_{1}$ = 2.25, $x_{2}$ = 3.5, $x_{3}$ = 4.75, $x_{4}$ = 6

$\int_{1}^{6}x^{4}$ dx  = $\frac{x^{5}}{5}$ = $\frac{7776 -1}{5}$
= 1555