Answer
$$
\int_{0}^{\frac{\pi}{2}} \sqrt[3] {1+\cos x}dx, \quad n=4
$$
(a) The approximation of the given integral by using the Trapezoidal Rule is $\approx 1.838967 $
(b) The approximation of the given integral by using the midpoints Rule is $\approx 1.845390 $
(c) The approximation of the given integral by using Simpson’s Rule is $\approx 1.843245 $
Work Step by Step
$$
\int_{0}^{\frac{\pi}{2}} \sqrt[3] {1+\cos x}dx, \quad n=4
$$
(a) Use the Trapezoidal Rule to approximate the given integral with the specified value of $n$.
With $ n =4, a = 0$, and $b = \frac{\pi}{2}$ we have
$$
\Delta x=\frac{b-a}{n}=\frac{\frac{\pi}{2}-0}{4}=\frac{\pi}{8}
$$ and so the Trapezoidal Rule gives:
$$
\begin{aligned} \int_{0}^{\frac{\pi}{2}} \sqrt[3] {1+\cos x}dx &= \frac{\Delta x}{2}\left[f\left(x_{0}\right)+2 f\left(x_{1}\right)+\cdots+2 f\left(x_{n-1}\right)+f\left(x_{n}\right)\right]\\
& \approx T_{4} \\
& =\frac{\pi}{8 \cdot 2}\left[f(0)+2 f\left(\frac{\pi}{8}\right)+2 f\left(\frac{2 \pi}{8}\right)+ \\
+2 f\left(\frac{3 \pi}{8}\right)+f\left(\frac{\pi}{2}\right)\right] \\ & \approx 1.838967
\end{aligned}
$$
(b) The midpoints Rule to approximate the given integral with the specified value of $ n=4 $ gives :
$$
\begin{aligned}\int_{0}^{\frac{\pi}{2}} \sqrt[3] {1+\cos x}dx &=\Delta x\left[f\left(\overline{x}_{1}\right)+f\left(\overline{x}_{2}\right)+\cdots+f\left(\overline{x}_{n}\right)\right] \\
& = M_{4} \\
& \approx
\frac{\pi}{8}\left[f\left(\frac{\pi}{16}\right)+f\left(\frac{3 \pi}{16}\right)+f\left(\frac{5 \pi}{16}\right)+f\left(\frac{7 \pi}{16}\right)\right]\\ & \approx 1.845390
\end{aligned}
$$
(c) Simpson’s Rule to approximate the given integral with the specified value of $ n=4$ gives :
$$
\begin{aligned} \int_{0}^{\frac{\pi}{2}} \sqrt[3] {1+\cos x}dx & = \frac{\Delta x}{3}\left[f\left(x_{0}\right)\right. +4 f\left(x_{1}\right)+2 f\left(x_{2}\right)+4 f\left(x_{3}\right)+\cdots \\ & \quad \left.+2 f\left(x_{n-2}\right)+4 f\left(x_{n-1}\right)+f\left(x_{n}\right)\right] \\
&= S_{4}\\
& \approx \frac{\pi}{8 \cdot 3}\left[f(0)+4 f\left(\frac{\pi}{8}\right)+2 f\left(\frac{2 \pi}{8}\right)+ \\
+4 f\left(\frac{3 \pi}{8}\right)+f\left(\frac{\pi}{2}\right)\right] \\
& \approx 1.843245
\end{aligned}
$$
