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

1. 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 approx 1.8961 )
(round to four decimal places as needed.)
an upper bound for ( left|e_{t}
ight| ) is
(round to four decimal places as needed.)

Explanation:

Step1: Recall the formula for the error bound in the trapezoidal rule

The error bound formula for the trapezoidal rule is \(|E_T|\leq\frac{(b - a)^3}{12n^2}M_2\), where \(a = 0\), \(b=\pi\), \(n = 4\), and \(M_2\) is the upper bound of \(|f''(t)|\) on \([a,b]\).
For \(y = f(t)=\sin t\), \(f'(t)=\cos t\), \(f''(t)=-\sin t\). On the interval \([0,\pi]\), \(|f''(t)|=|-\sin t|\leq1\), so \(M_2 = 1\).

Step2: Substitute the values into the error - bound formula

Substitute \(a = 0\), \(b=\pi\), \(n = 4\), and \(M_2 = 1\) into the formula \(|E_T|\leq\frac{(b - a)^3}{12n^2}M_2\).
We have \(\frac{(\pi-0)^3}{12\times4^2}\times1=\frac{\pi^3}{12\times16}\).
Calculate \(\frac{\pi^3}{192}\approx\frac{31.00627668}{192}\approx0.1615\).

Answer:

\(0.1615\)