Question

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]?

Answer 1

See the explanation section, below.

Explanation:

For this question we have `f(x) = 9sqrt(lnx)`

`[a,b] = [1,4]` and `n=6`

For all three approximations, we have

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

(To eveluate `f(x)`, 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 `Deltax = 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 = 1/2 Deltax[f(x_0)+2f(x_1)+2f(x_2)+2f(x_3)+2f(x_4)+2f(x_5)+f(x_6)]`

Simpson's:
`S_6 = 1/3 Deltax[f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+2f(x_4)+4f(x_5)+f(x_6)]`

Do the arithmetic.

Midpoint rule , needs the midpoints of the subintervals. We'll call them `m_i`. Where `m_i = 1/2(x_(i-1)+x_i)`.

`m_1=1.25, m_2=1.75, m_3=2.25, m_4=2.75, m_5=3.25, m_6=3.75`

`Mid_6 = Deltax[f(m_1)+f(m_2)+f(m_3)+f(m_4)+f(m_5)+f(m_6)]`

Do the arithmetic.