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\n$$ int _ { - pi / 2 } ^ { pi / 2 } 5 cos t d t $$\n\n1. 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\n( t approx 9.4806 )\n(round to four decimal places as needed.)\n\nan upper bound for ( left| e _ { t } \right| ) is \n(round to four decimal places as needed.)

the instructions for the given integral have two parts, one for the trapezoidal rule and one for simpsons rule. complete the following parts\n\n$$ int _ { - pi / 2 } ^ { pi / 2 } 5 cos t d t $$\n\n1. 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\n( t approx 9.4806 )\n(round to four decimal places as needed.)\n\nan upper bound for ( left| e _ { t } \right| ) is \n(round to four decimal places as needed.)

Answer

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), where (a=-\frac{\pi}{2}), (b = \frac{\pi}{2}), (n = 4), and (M) is the upper bound of (|f''(t)|) on the interval ([a,b]).

Given (f(t)=5\cos t), then (f'(t)=- 5\sin t) and (f''(t)=-5\cos t). The absolute value (|f''(t)| = 5|\cos t|). On the interval (\left[-\frac{\pi}{2},\frac{\pi}{2}\right]), the maximum value of (|\cos t|) is (1). So (M = 5).

Step2: Substitute the values into the error - bound formula

Substitute (a=-\frac{\pi}{2}), (b=\frac{\pi}{2}), (n = 4), and (M = 5) into the formula (|E_T|\leq\frac{(b - a)^3}{12n^2}M).

First, calculate (b - a=\frac{\pi}{2}-\left(-\frac{\pi}{2}\right)=\pi).

Then (\frac{(b - a)^3}{12n^2}M=\frac{\pi^3}{12\times4^2}\times5).

[ \begin{align*} \frac{\pi^3\times5}{12\times16}&=\frac{5\pi^3}{192}\ &\approx\frac{5\times31.00627668}{192}\ &=\frac{155.0313834}{192}\ &\approx0.807455 \end{align*} ]

Answer:

(0.8075)