Question

How do you use the Trapezoidal Rule with n=4 to approximate from [2,3] of `1/(x-1)^2 dx`?

Answer 1

Approximate the Integral `int_a^b f(x) dx` using trapezoidal approximation with `n` intervals.

In this question we have:
`f(x) = 1/(x-1)^2`
`{a,b] = [2, 3]`, and
`n=4`.

So we get
`Delta x = (b-a)/n = (3-2)/4 = 1/4 = 0.25`

The endpoints of the subintervals are found by beginning at `a=2` and successively adding `Delta x = 1/4` to find the points until we get to `x_n = b = 3`.

`x_0 = 2`, `x_1 = 9/4`, `x_2 = 10/4 = 5/2`, `x_3 = 11/4`, and `x_4 = 12/4 = 3 = b`

Now apply the formula (do the arithmetic) for `f(x) = 1/(x-1)^2`

`T_4=(Deltax)/2 [f(x_0)+2f(x_1)+2f(x_2)+ * * * 2f(x_9)+f(x_10)]`