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]?

1 Answer
Oct 11, 2015

Answer:

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.