# How do you use the Trapezoidal Rule with step size n=4 to estimate int t^3 +t) dx with [0,2]?

Nov 17, 2015

Assuming that the variables should match see below.

#### Explanation:

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

In this question we have:
$f \left(x\right) = {x}^{3} + x$
$\left\{a , b\right] = \left[0 , 2\right]$, and
$n = 4$.

So we get
$\Delta x = \frac{b - a}{n} = \frac{2 - 0}{4} = \frac{1}{2}$

The endpoints of the subintervals are found by beginning at $a = 0$ and successively adding $\Delta x = \frac{1}{2}$ to find the points until we get to ${x}_{n} = b = 4$.

${x}_{0} = 0$, ${x}_{1} = \frac{1}{2}$, ${x}_{2} = \frac{2}{2} = 1$, ${x}_{3} = \frac{3}{2}$ and ${x}_{4} = \frac{4}{2} = 2 = b$

Now apply the formula (do the arithmetic) for $f \left(x\right) = {x}^{3} + x$.

${T}_{4} = \frac{1}{2} \Delta x \left[f \left({x}_{0}\right) + 2 f \left({x}_{1}\right) + 2 f \left({x}_{2}\right) + 2 f \left({x}_{3}\right) + f \left({x}_{4}\right)\right]$