Find table lists several measurements gathered in an experiment to approximate an unknown continuous function $y = f ( x )$. Approximate the integral $\int _ { 0 } ^ { 2 } f ( x ) d x$ using the Simpson's Rule. Round your answer to three decimal places.
$\begin{array}{|l|l|l|l|l|l|l|l|l|l|}
\hline x & 0.00 & 0.25 & 0.50 & 0.75 & 1.00 & 1.25 & 1.50 & 1.75 & 2.00 \
\hline y & 4.72 & 4.77 & 4.82 & 6.17 & 6.52 & 7.17 & 7.32 & 7.97 & 8.02 \
\hline
\end{array}$

A) $12.197$
B) $8.518$
C) $15.975$
D) $12.865$
E) $11.628$

Question

Find table lists several measurements gathered in an experiment to approximate an unknown continuous function $y = f ( x )$. Approximate the integral $\int _ { 0 } ^ { 2 } f ( x ) d x$ using the Simpson's Rule. Round your answer to three decimal places.
$\begin{array}{|l|l|l|l|l|l|l|l|l|l|}
\hline x & 0.00 & 0.25 & 0.50 & 0.75 & 1.00 & 1.25 & 1.50 & 1.75 & 2.00 \
\hline y & 4.72 & 4.77 & 4.82 & 6.17 & 6.52 & 7.17 & 7.32 & 7.97 & 8.02 \
\hline
\end{array}$

A) $12.197$
B) $8.518$
C) $15.975$
D) $12.865$
E) $11.628$

Quizplus · Accepted Answer

Simpson's Rule for approximating integrals, especially when the interval is divided into an even number of segments, is given by $\int_a^b f(x) dx \approx \frac{\Delta x}{3} [f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \cdots + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n)]$, where $\Delta x$ is the width of each segment. In this case, $\Delta x = 0.25$ and there are 9 intervals (8 segments), fitting the requirement for Simpson's Rule. Applying the rule to the given data yields an approximate integral value of $12.865$, after calculating and rounding to three decimal places.

Find Table Lists Several Measurements Gathered in an Experiment to Approximate