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

Oct 27, 2015

Approximate the Integral ${\int}_{a}^{b} f \left(x\right) \mathrm{dx}$ using trapezoidal approximation with $n$ intervals.

In this question we have:
$f \left(x\right) = \frac{1}{x - 1} ^ 2$
$\left\{a , b\right] = \left[2 , 3\right]$, and
$n = 4$.

So we get
$\Delta x = \frac{b - a}{n} = \frac{3 - 2}{4} = \frac{1}{4} = 0.25$

The endpoints of the subintervals are found by beginning at $a = 2$ and successively adding $\Delta x = \frac{1}{4}$ to find the points until we get to ${x}_{n} = b = 3$.

${x}_{0} = 2$, ${x}_{1} = \frac{9}{4}$, ${x}_{2} = \frac{10}{4} = \frac{5}{2}$, ${x}_{3} = \frac{11}{4}$, and ${x}_{4} = \frac{12}{4} = 3 = b$

Now apply the formula (do the arithmetic) for $f \left(x\right) = \frac{1}{x - 1} ^ 2$

${T}_{4} = \frac{\Delta x}{2} \left[f \left({x}_{0}\right) + 2 f \left({x}_{1}\right) + 2 f \left({x}_{2}\right) + \cdot \cdot \cdot 2 f \left({x}_{9}\right) + f \left({x}_{10}\right)\right]$