Question

theinstructions for the given integral have two parts, one for the trapezoidal rule and one for simpsons rule. complete the following parts.
int_{0}^{2}left(3 t^{3}+5 t
ight) d t
i. using the trapezoidal rule complete the following.
a. estimate the integral with ( n = 4 ) steps and find an upper bound for ( left|e_{t}
ight| ).
( t=square )
(simplify your answer.)

Explanation:

Step1: Calculate \(\Delta t\)

The formula for \(\Delta t=\frac{b - a}{n}\). Here, \(a = 0\), \(b=2\), \(n = 4\). So \(\Delta t=\frac{2-0}{4}=0.5\).

Step2: Find the endpoints \(t_i\)

\(t_0=a = 0\), \(t_1=t_0+\Delta t=0.5\), \(t_2=t_1+\Delta t = 1\), \(t_3=t_2+\Delta t=1.5\), \(t_4=t_3+\Delta t = 2\).

Step3: Calculate \(f(t_i)\)

\(f(t)=3t^{3}+5t\).
\(f(t_0)=f(0)=3\times0^{3}+5\times0 = 0\).
\(f(t_1)=f(0.5)=3\times(0.5)^{3}+5\times(0.5)=3\times0.125 + 2.5=0.375+2.5 = 2.875\).
\(f(t_2)=f(1)=3\times1^{3}+5\times1=3 + 5=8\).
\(f(t_3)=f(1.5)=3\times(1.5)^{3}+5\times(1.5)=3\times3.375+7.5 = 10.125+7.5=17.625\).
\(f(t_4)=f(2)=3\times2^{3}+5\times2=3\times8 + 10=24 + 10=34\).

Step4: Apply the trapezoidal rule formula \(T=\frac{\Delta t}{2}[f(t_0)+2f(t_1)+2f(t_2)+2f(t_3)+f(t_4)]\)

\(T=\frac{0.5}{2}[0 + 2\times2.875+2\times8+2\times17.625+34]\)
\(=\frac{0.5}{2}(0 + 5.75+16+35.25+34)\)
\(=\frac{0.5}{2}(91)\)
\(T = 22.75\)

Answer:

\(22.75\)