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}^{\pi} \sin t d t \\)
\\( t \approx 1.8961 \\)
(round to four decimal places as needed.)
an upper bound for \\( \left|e_{t}\
ight| \\) is 0.1615.
(round to four decimal places as needed.)
b. evaluate the integral directly and find \\( \left|e_{t}\
ight| \\).
\\( \int_{0}^{\pi} \sin d t=2 \\)
(type an exact answer in simplified form.)
\\( \left|e_{t}\
ight|=\square \\)
(round to four decimal places as needed.)

Explanation:

Step1: Recall the formula for \(|E_T|\)

\(|E_T|=\vert\text{Exact Value}-\text{Trapezoidal Approximation}\vert\)

Step2: Substitute the values

The exact value of \(\int_{0}^{\pi}\sin tdt = 2\) and the trapezoidal approximation \(T\approx1.8961\)
\(|E_T|=\vert2 - 1.8961\vert\)

Step3: Calculate the result

\(|E_T|=2- 1.8961=0.1039\)

Answer:

\(0.1039\)