Question

How do you use the trapezoidal rule with n=6 to approximate the area between the curve `6sqrt(lnx)` from 1 to 4?

Answer 1

See the explanation section, below.

Explanation:

To approximate the Integral `int_a^b f(x) dx` using trapezoidal approximation with `n` intervals, use
`T_n=(Deltax)/2 [f(x_0)+2f(x_1)+2f(x_2)+ * * * 2f(x_(n-1))+f(x_n)]`

In this question we have:
`f(x) = 6sqrt(lnx)`
`{a,b] = [1, 4]`, and
`n=6`.

So we get
`Delta x = (b-a)/n = (4-1)/6 = 1/2 = 0.5`

The endpoints of the subintervals are found by beginning at `a=1` and successively adding `Delta x = 0.5` to find the points until we get to `x_n = b = 4`.

`x_0 = 1`, `x_1 = 1.5`, `x_2 = 2`, `x_3 = 2.5`, `x_4 = 3`, `x_5 = 3.5`, `x_6 = 4=b`,

Now apply the formula (do the arithmetic) for `f(x) = 6sqrt(lnx)`.

`T_6=(Deltax)/2 [f(x_0)+2f(x_1)+2f(x_2)+ * * * 2f(x_5)+f(x_6)]`