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.

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.

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Answer 2

To approximate the integral (\int_{1}^{4} \sqrt{\ln x} , dx) using the Trapezoidal Rule, Midpoint Rule, and Simpson's Rule with (n = 6), follow these steps:

  1. Trapezoidal Rule:

    [h = \frac{b-a}{n} = \frac{4-1}{6} = \frac{1}{2}]

    [x_0 = 1, x_1 = 1.5, x_2 = 2, x_3 = 2.5, x_4 = 3, x_5 = 3.5, x_6 = 4]

    [T_n = \frac{h}{2}[f(x_0) + 2\sum_{i=1}^{n-1}f(x_i) + f(x_n)]]

  2. Midpoint Rule:

    [m_i = \frac{x_{i-1} + x_i}{2}]

    [M_n = h\sum_{i=1}^{n}f(m_i)]

  3. Simpson's Rule:

    [S_n = \frac{h}{3}[f(x_0) + 4\sum_{i=1}^{n/2}f(x_{2i-1}) + 2\sum_{i=1}^{n/2-1}f(x_{2i}) + f(x_n)]]

Now, plug in the values and calculate the approximations.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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