selected values from the function f(x) are shown in the table below. when a trapezoidal sum with two…
selected values from the function f(x) are shown in the table below. when a trapezoidal sum with two subintervals is used to approximate ∫₂⁵ f(x)dx, the value is x 2 4 5 f(x) 5 1 5 9 7 11 3
Answer
Explanation:
Step1: Recall trapezoidal - sum formula
The trapezoidal - sum formula for approximating $\int_{a}^{b}f(x)dx$ with $n$ sub - intervals is $T_n=\frac{\Delta x}{2}[f(x_0)+2f(x_1)+\cdots + 2f(x_{n - 1})+f(x_n)]$, where $\Delta x=\frac{b - a}{n}$. Here, $a = 2$, $b = 5$, and $n = 2$. Then $\Delta x=\frac{5 - 2}{2}=\frac{3}{2}$. Also, $x_0 = 2$, $x_1 = 4$, $x_2 = 5$, $f(x_0)=5$, $f(x_1)=1$, $f(x_2)=5$.
Step2: Apply the formula
$T_2=\frac{\Delta x}{2}[f(x_0)+2f(x_1)+f(x_2)]$. Substitute $\Delta x=\frac{3}{2}$, $f(x_0)=5$, $f(x_1)=1$, and $f(x_2)=5$ into the formula: [ \begin{align*} T_2&=\frac{\frac{3}{2}}{2}(5 + 2\times1+5)\ &=\frac{3}{4}(5 + 2+5)\ &=\frac{3}{4}(12)\ &=9 \end{align*} ]
Answer:
9