Use Simpson's Rule with n = 4 steps to estimate the integral.
-$$\int _ { - \pi } ^ { 0 } \sin x d x$$
A) $- \frac { 1 + 2 \sqrt { 2 } } { 6 } \pi$
B) $- ( 1 + 2 \sqrt { 2 } ) \pi$
C) $- \frac { 1 + \sqrt { 2 } } { 4 } \pi$
D) $- \frac { 2 + \sqrt { 2 } } { 6 } \pi$

Question

Quizplus · Accepted Answer

Simpson's Rule formula for approximating integrals is given by $\frac{\Delta x}{3} [f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + f(x_4)]$, where $\Delta x = \frac{b - a}{n}$ and $n$ is the number of intervals. For the integral $\int _ { - \pi } ^ { 0 } \sin x d x$ with $n = 4$, we have $a = -\pi$, $b = 0$, and $\Delta x = \frac{\pi - (-\pi)}{4} = \frac{\pi}{2}$. The x-values are: $x_0 = -\pi$, $x_1 = -\frac{\pi}{2}$, $x_2 = 0$, $x_3 = \frac{\pi}{2}$, and $x_4 = \pi$. Substituting these into the formula and simplifying gives $- \frac { 1 + 2 \sqrt { 2 } } { 6 } \pi$.

Use Simpson's Rule with N = 4 Steps to Estimate ∫−π0sin⁡xdx\int _ { - \pi } ^ { 0 } \sin x d x∫−π0​sinxdx

Use Simpson's Rule with N = 4 Steps to Estimate $\int _ { - \pi } ^ { 0 } \sin x d x$