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

\(0.75\)