Question

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

Answer 1

Assuming that the variables should match see below.

Explanation:

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

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

So we get
`Delta x = (b-a)/n = (2-0)/4 = 1/2`

The endpoints of the subintervals are found by beginning at `a=0` and successively adding `Delta x = 1/2` to find the points until we get to `x_n = b = 4`.

`x_0 = 0`, `x_1 = 1/2`, `x_2 =2/2 = 1`, `x_3 = 3/2` and `x_4 = 4/2 = 2 = b`

Now apply the formula (do the arithmetic) for `f(x) = x^3+x`.

`T_4=1/2Deltax [f(x_0)+2f(x_1)+2f(x_2)+2f(x_3) + f(x_4)]`