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 Trapezoidal Rule. Round your answer to three decimal places.
$\begin{array}{|l|l|l|l|l|l|l|l|l|l|}
\hline x & 0.00 & 0.25 & 0.50 & 0.75 & 1.00 & 1.25 & 1.50 & 1.75 & 2.00 \
\hline y & 4.12 & 4.27 & 4.62 & 6.37 & 6.82 & 7.47 & 7.72 & 8.67 & 8.92 \
\hline
\end{array}$

A) $13.115$
B) $12.000$
C) $7.373$
D) $104.920$
E) $24.600$

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 Trapezoidal Rule. Round your answer to three decimal places.
$\begin{array}{|l|l|l|l|l|l|l|l|l|l|}
\hline x & 0.00 & 0.25 & 0.50 & 0.75 & 1.00 & 1.25 & 1.50 & 1.75 & 2.00 \
\hline y & 4.12 & 4.27 & 4.62 & 6.37 & 6.82 & 7.47 & 7.72 & 8.67 & 8.92 \
\hline
\end{array}$

A) $13.115$
B) $12.000$
C) $7.373$
D) $104.920$
E) $24.600$

Quizplus · Accepted Answer

The Trapezoidal Rule formula for approximating the integral of a function $f(x)$ over an interval $[a, b]$ with $n$ subintervals of equal width $\Delta x = \frac{b-a}{n}$ is given by:$$\int_a^b f(x)dx \approx \frac{\Delta x}{2} [f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n)]$$Given the table, we have $n = 8$ intervals (from $x = 0$ to $x = 2$ with a step of $0.25$), so $\Delta x = 0.25$. Applying the formula:$$\int_0^2 f(x)dx \approx \frac{0.25}{2} [4.12 + 2(4.27 + 4.62 + 6.37 + 6.82 + 7.47 + 7.72) + 8.67 + 8.92] = 13.115$$

Find Table Lists Several Measurements Gathered in an Experiment to Approximate