$\text { Use Simpson's Rule to approximate the value of the definite integral } \int _ { 0 } ^ { 8 } 2 \sqrt [ 4 ] { x } d x \text { with }$ $n = 8$. Round your answer to four decimal places
A) $16.9283$
B) $22.3017$
C) $21.1805$
D) $24.5441$
E) $18.5139$

Question

Quizplus · Accepted Answer

First, we need to calculate the step size:

$$h = \frac{b-a}{n}=\frac{8-0}{8}=1$$

Then, we can use Simpson's Rule:

$$\int _ { 0 } ^ { 8 } 2 \sqrt [ 4 ] { x } d x \approx \frac{h}{3} \left[f(a) + 4f(a+h) + 2f(a+2h) + 4f(a+3h) + 2f(a+4h) + 4f(a+5h) + 2f(a+6h) + f(b)ight]$$

where $f(x) = 2\sqrt[4]{x}$.

Plugging in the values, we get:

\begin{align*}
\int _ { 0 } ^ { 8 } 2 \sqrt [ 4 ] { x } d x &\approx \frac{1}{3} \left[2\sqrt[4]{0} + 4\cdot 2\sqrt[4]{1} + 2\cdot 2\sqrt[4]{2} + 4\cdot 2\sqrt[4]{3} + 2\cdot 2\sqrt[4]{4} ight.\
&\qquad\left. + 4\cdot 2\sqrt[4]{5} + 2\cdot 2\sqrt[4]{6} + 2\sqrt[4]{8}ight]\
&= 21.1805\ ( 	ext{rounded to four decimal places})
\end{align*}

Therefore, the best choice is C.