the instructions for the given integral have two parts, one for the trapezoidal rule and one for simpsons…

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

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

Answer

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)