How do you use the Trapezoidal Rule to approximate integral #int(2/x) dx# for n=4 from [1,3]?
# int_1^3 \ 2/x \ dx ~~ 2.233333#
We have:
# y = 2/x #
We want to estimate
# Deltax = (31)/4 = 0.5#
The values of the function are tabulated as follows;
Trapezium Rule
# A = int_a^b \ y \ dx #
# \ \ \ ~~ h/2{y_0+y_n+2(y_1+...+y_(n1)) } #
# \ \ \ = 0.5/2 * { 2 + 0.666667 + 2*(1.333333 + 1 + 0.8) } #
# \ \ \ = 0.25 * { 2.666667 + 2*(3.133333) }#
# \ \ \ = 0.25 * { 2.666667 + 6.266667 }#
# \ \ \ = 0.25 * 8.933333#
# \ \ \ = 2.233333#
Actual Value
For comparison of accuracy:
# A = int_1^3 \ 2/x \ dx #
# \ \ \ = [2lnx]_1^3 #
# \ \ \ = 2ln32ln1 #
# \ \ \ = 2ln3 #
# \ \ \ ~~ 2.1972 #
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To approximate the integral ( \int_{1}^{3} \frac{2}{x} , dx ) using the Trapezoidal Rule with ( n = 4 ), follow these steps:

Divide the interval ([1, 3]) into ( n ) subintervals of equal width. Since ( n = 4 ), each subinterval will have width ( \Delta x = \frac{3  1}{4} = 0.5 ).

Determine the function values at the endpoints of each subinterval. For ( n = 4 ), these points are ( x_0 = 1 ), ( x_1 = 1.5 ), ( x_2 = 2 ), ( x_3 = 2.5 ), and ( x_4 = 3 ).

Use the Trapezoidal Rule formula:
[ \int_{a}^{b} f(x) , dx \approx \frac{\Delta x}{2} \left[ f(x_0) + 2f(x_1) + 2f(x_2) + 2f(x_3) + f(x_4) \right] ]
Substitute the values:
[ \int_{1}^{3} \frac{2}{x} , dx \approx \frac{0.5}{2} \left[ \frac{2}{1} + 2\left(\frac{2}{1.5}\right) + 2\left(\frac{2}{2}\right) + 2\left(\frac{2}{2.5}\right) + \frac{2}{3} \right] ]
 Calculate the approximate value:
[ \int_{1}^{3} \frac{2}{x} , dx \approx \frac{1}{4} \left[ 2 + \frac{8}{1.5} + 4 + \frac{8}{2.5} + \frac{2}{3} \right] ]
[ = \frac{1}{4} \left[ 2 + \frac{16}{3} + 4 + \frac{16}{5} + \frac{2}{3} \right] ]
[ = \frac{1}{4} \left[ 2 + \frac{32}{15} + \frac{20}{5} + \frac{2}{3} \right] ]
[ = \frac{1}{4} \left[ 2 + \frac{32}{15} + 4 + \frac{2}{3} \right] ]
[ = \frac{1}{4} \left[ 6 + \frac{32}{15} + \frac{2}{3} \right] ]
[ = \frac{1}{4} \left[ \frac{90 + 32 + 10}{15} \right] ]
[ = \frac{1}{4} \left[ \frac{132}{15} \right] ]
[ = \frac{33}{15} ]
[ = 2.2 ]
So, the approximate value of ( \int_{1}^{3} \frac{2}{x} , dx ) using the Trapezoidal Rule with ( n = 4 ) is ( 2.2 ).
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