use simpsons rule and all the data in the following table to estimate the value of the integral ∫₂₀²⁶ ydx. x…

Explanation:

Step1: Identify the values of $n$, $a$, $b$ and $\Delta x$

$n = 6$ (number of sub - intervals), $a = 20$, $b = 26$, $\Delta x=\frac{b - a}{n}=\frac{26 - 20}{6}=1$.

Step2: Recall Simpson's Rule formula

Simpson's Rule for $\int_{a}^{b}y\mathrm{d}x\approx\frac{\Delta x}{3}(y_0 + 4y_1+2y_2 + 4y_3+2y_4 + 4y_5+y_6)$.

Step3: Identify the $y$ - values

$y_0 = 2$, $y_1=-2$, $y_2 = 3$, $y_3=-4$, $y_4=-3$, $y_5=-3$, $y_6=-7$.

Step4: Substitute values into the formula

$\int_{20}^{26}y\mathrm{d}x\approx\frac{1}{3}(2+4\times(-2)+2\times3 + 4\times(-4)+2\times(-3)+4\times(-3)+(-7))$. First, calculate each term: $4\times(-2)=-8$, $2\times3 = 6$, $4\times(-4)=-16$, $2\times(-3)=-6$, $4\times(-3)=-12$. Then, $\frac{1}{3}(2-8 + 6-16-6-12-7)=\frac{1}{3}(2 + 6-(8 + 16+6+12+7))=\frac{1}{3}(8 - 49)=-\frac{41}{3}\approx - 13.6667$.

Answer:

$-13.6667$