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

To use the trapezoidal rule with $n = 6$ to approximate the area between the curve $6\sqrt{\ln(x)}$ from $1$ to $4$, follow these steps:

1. Divide the interval $[1, 4]$ into $n = 6$ equal subintervals.
2. Calculate the width of each subinterval, $\Delta x$, using the formula: $\Delta x = \frac{b - a}{n}$, where $a = 1$ and $b = 4$.
3. Evaluate the function $6\sqrt{\ln(x)}$ at each endpoint of the subintervals and at the midpoint of each subinterval.
4. Apply the trapezoidal rule formula: $A \approx \frac{\Delta x}{2}\left[f(x_0) + 2\sum_{i=1}^{n-1}f(x_i) + f(x_n)\right]$, where $x_0$ is the lower limit of integration, $x_n$ is the upper limit of integration, and $x_i$ are the midpoints of the subintervals.
5. Substitute the values of $f(x_i)$ into the formula and calculate the sum to obtain the approximate area.

Using $n = 6$, the subintervals are: $[1, 1.5]$, $[1.5, 2]$, $[2, 2.5]$, $[2.5, 3]$, $[3, 3.5]$, and $[3.5, 4]$.

The width of each subinterval is $\Delta x = \frac{4 - 1}{6} = \frac{1}{2}$.

Evaluate the function at the endpoints and midpoints of the subintervals:

• $f(1) = 6\sqrt{\ln(1)} = 0$
• $f(1.5)$
• $f(2)$
• $f(2.5)$
• $f(3)$
• $f(3.5)$
• $f(4)$

Using these values, apply the trapezoidal rule formula to approximate the area.