# How do you use the Trapezoidal Rule, the Midpoint Rule, and Simpson's Rule to approximate the given integral with the specified value of n=6 for int 9 sqrt (ln x) dx from [1,4]?

Oct 11, 2015

See the explanation section, below.

#### Explanation:

For this question we have $f \left(x\right) = 9 \sqrt{\ln x}$

$\left[a , b\right] = \left[1 , 4\right]$ and $n = 6$

For all three approximations, we have

$\Delta x = \frac{b - a}{n} = \frac{4 - 1}{6} = \frac{1}{2} = 0.5$

(To eveluate $f \left(x\right)$, we'll want a calculator or tables, so decimals are preferable to fractions for this problem.)

We need the endpoints of the 6 subintervals. Start at $a = 1$ and successively add $\Delta x = 0.5$ until we get to $b = 4$.

The endpoints of the subintervals are:

${x}_{0} = 1 , {x}_{1} = 1.5 , {x}_{2} = 2 , {x}_{3} = 2.5 , {x}_{4} = 3 , {x}_{5} = 3.5 , {x}_{6} = 4$

Now for the Trapezoidal and Simpson's Rules just apply the formula and do the arithmetic
Trapezoidal:
${T}_{6} = \frac{1}{2} \Delta x \left[f \left({x}_{0}\right) + 2 f \left({x}_{1}\right) + 2 f \left({x}_{2}\right) + 2 f \left({x}_{3}\right) + 2 f \left({x}_{4}\right) + 2 f \left({x}_{5}\right) + f \left({x}_{6}\right)\right]$

Simpson's:
${S}_{6} = \frac{1}{3} \Delta x \left[f \left({x}_{0}\right) + 4 f \left({x}_{1}\right) + 2 f \left({x}_{2}\right) + 4 f \left({x}_{3}\right) + 2 f \left({x}_{4}\right) + 4 f \left({x}_{5}\right) + f \left({x}_{6}\right)\right]$

Do the arithmetic.

Midpoint rule , needs the midpoints of the subintervals. We'll call them ${m}_{i}$. Where ${m}_{i} = \frac{1}{2} \left({x}_{i - 1} + {x}_{i}\right)$.

${m}_{1} = 1.25 , {m}_{2} = 1.75 , {m}_{3} = 2.25 , {m}_{4} = 2.75 , {m}_{5} = 3.25 , {m}_{6} = 3.75$

$M i {d}_{6} = \Delta x \left[f \left({m}_{1}\right) + f \left({m}_{2}\right) + f \left({m}_{3}\right) + f \left({m}_{4}\right) + f \left({m}_{5}\right) + f \left({m}_{6}\right)\right]$

Do the arithmetic.