Question

How do you use the trapezoidal rule with n=4 to approximate the area between the curve `y=x^2+4x` from 0 to 4?

Answer 1

See the explanation below.

Explanation:

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

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

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

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

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

Now apply the formula (do the arithmetic) for `f(x) = x^2+4x`.

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