aflacatn

2020-12-12

Use the Composite Trapezoidal, Simpson's and Midpoint Rules to approximate the integral

(For the Midpoint Rule use

gwibdaithq

Skilled2020-12-13Added 84 answers

a) Trapezoidal Rule:

The interval [1, 2 can be partitioned into 4 sub-intervals,$P=\{1,\text{}\frac{5}{4},\text{}\frac{3}{2},\text{}\frac{7}{4},\text{}2\},$ where:

$\mathrm{\Delta}\text{}x=\frac{b\text{}-\text{}a}{4}=\frac{1}{4}.$

${\int}_{1}^{2}\text{}x\mathrm{ln}(x)dx=\frac{1}{4}\{f(1)\text{}+\text{}2f\left(\frac{5}{4}\right)\text{}+\text{}2f\left(\frac{3}{2}\right)\text{}+\text{}2f\left(\frac{7}{4}\right)\text{}+\text{}f(2)\}$

$\frac{1}{4}\{(\mathrm{ln}(1))\text{}+\text{}2(\frac{5}{4}\text{}\mathrm{ln}\left(\frac{5}{4}\right))\text{}+\text{}2(\frac{3}{2}\mathrm{ln}\left(\frac{3}{2}\right))\text{}+\text{}2(\frac{7}{4}\mathrm{ln}\left(\frac{7}{4}\right))\text{}+\text{}2(2\mathrm{ln}(2))\}$

$\frac{1}{4}\{0\text{}+\text{}0.55788589\text{}+\text{}1.2163953\text{}+\text{}1.9586553\text{}+\text{}2.7725887\}=1.6263813$

b) Simpsons

The interval [1, 2 can be partitioned into 4 sub-intervals,

b) Simpsons

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