Question

How do you use the Trapezoidal Rule to approximate integral `int(2/x) dx` for n=4 from [1,3]?

Answer 1

`int_1^3 \ 2/x \ dx ~~ 2.233333`

Explanation:

We have:

`y = 2/x`

We want to estimate `int \ y \ dx` over the interval `[1,3]` with `4` strips; thus:

`Deltax = (3-1)/4 = 0.5`

The values of the function are tabulated as follows;

Trapezium Rule

`A = int_a^b \ y \ dx`

`\ \ \ ~~ h/2{y_0+y_n+2(y_1+...+y_(n-1)) }`

`\ \ \ = 0.5/2 * { 2 + 0.666667 + 2*(1.333333 + 1 + 0.8) }`
`\ \ \ = 0.25 * { 2.666667 + 2*(3.133333) }`
`\ \ \ = 0.25 * { 2.666667 + 6.266667 }`
`\ \ \ = 0.25 * 8.933333`
`\ \ \ = 2.233333`

Actual Value

For comparison of accuracy:

`A = int_1^3 \ 2/x \ dx`
`\ \ \ = [2lnx]_1^3`
`\ \ \ = 2ln3-2ln1`
`\ \ \ = 2ln3`
`\ \ \ ~~ 2.1972`