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

Question

Aurora Alvarado · Accepted Answer

See the explanation section, below.

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)] #

Paisley Johnson · Answer

undefined 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.