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 _ { 0 } ^ { 2 } \\left( 3 t ^ { 3 } + 5 t \\right) d t \\)\n\\( t = 22.75 \\)\n(simplify your answer.)\nthe upper bound for \\( \\left| e _ { t } \\right| \\) is 1.5.\n(simplify your answer.)\nb. evaluate the integral directly and find \\( \\left| e _ { t } \\right| \\).\n\\( \\int _ { 0 } ^ { 2 } \\left( 3 t ^ { 3 } + 5 t \\right) d t = 22 \\)\n(simplify your answer.)\n\\( \\left| e _ { t } \\right| = \\square \\)\n(simplify your answer.)

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 _ { 0 } ^ { 2 } \\left( 3 t ^ { 3 } + 5 t \\right) d t \\)\n\\( t = 22.75 \\)\n(simplify your answer.)\nthe upper bound for \\( \\left| e _ { t } \\right| \\) is 1.5.\n(simplify your answer.)\nb. evaluate the integral directly and find \\( \\left| e _ { t } \\right| \\).\n\\( \\int _ { 0 } ^ { 2 } \\left( 3 t ^ { 3 } + 5 t \\right) d t = 22 \\)\n(simplify your answer.)\n\\( \\left| e _ { t } \\right| = \\square \\)\n(simplify your answer.)

Answer

Explanation:

Step1: Recall the formula for (|E_T|)

The formula for the error in the trapezoidal rule is (|E_T|=|T - I|), where (T) is the trapezoidal - rule approximation and (I) is the exact value of the integral.

Step2: Substitute the known values

We are given that (T = 22.75) and (I=\int_{0}^{2}(3t^{3}+5t)dt = 22). Substitute these values into the formula (|E_T|=|22.75 - 22|).

Step3: Calculate the absolute value

(|E_T|=|22.75 - 22|=|0.75| = 0.75)

Answer:

(0.75)