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

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

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

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)