Question

the instructions for the given integral have two parts, one for the trapezoidal rule and one for simpsons rule. complete the following parts.
int_{3}^{13} \frac{1}{s^{2}} d s
i. using the trapezoidal rule
a. estimate the integral with ( n = 4 ) steps and find an upper bound for ( left|e_{t}
ight| ).
the estimate using the trapezoidal rule with ( n = 4 ) is
(round to four decimal places as needed.)

Explanation:

Step1: Calculate \(\Delta s\)

The formula for \(\Delta s=\frac{b - a}{n}\), where \(a = 3\), \(b = 13\), \(n=4\).
\(\Delta s=\frac{13 - 3}{4}=\frac{10}{4} = 2.5\)

Step2: Find the sub - intervals

The sub - intervals are \(s_0=3\), \(s_1=3 + 2.5=5.5\), \(s_2=5.5+2.5 = 8\), \(s_3=8 + 2.5=10.5\), \(s_4=10.5+2.5 = 13\)

Step3: Calculate \(f(s)\) at each sub - interval

\(f(s)=\frac{1}{s^{2}}\)
\(f(s_0)=\frac{1}{3^{2}}=\frac{1}{9}\approx0.1111\)
\(f(s_1)=\frac{1}{5.5^{2}}=\frac{1}{30.25}\approx0.0330\)
\(f(s_2)=\frac{1}{8^{2}}=\frac{1}{64}\approx0.0156\)
\(f(s_3)=\frac{1}{10.5^{2}}=\frac{1}{110.25}\approx0.0091\)
\(f(s_4)=\frac{1}{13^{2}}=\frac{1}{169}\approx0.0059\)

Step4: Apply the trapezoidal rule formula \(T_n=\frac{\Delta s}{2}[f(s_0)+2f(s_1)+2f(s_2)+2f(s_3)+f(s_4)]\)

\(T_4=\frac{2.5}{2}[0.1111 + 2\times0.0330+2\times0.0156+2\times0.0091+0.0059]\)
\(=\frac{2.5}{2}(0.1111+0.0660 + 0.0312+0.0182+0.0059)\)
\(=\frac{2.5}{2}(0.2324)\)
\(=2.5\times0.1162\)
\(=0.2905\)

Answer:

\(0.2905\)