Use the Trapezoidal Rule to approximate the integral with answers rounded to four decimal places. $$\int _ { 0 } ^ { 1 } \frac { d x } { 2 x + 4 } ; \quad n = 7$$
A) $0.2029$
B) $0.1088$
C) $0.1163$
D) $0.2326$

Question

Quizplus · Accepted Answer

The Trapezoidal Rule is given by$$T_n=\frac{b-a}{2n}[f(x_0)+2f(x_1)+2f(x_2)+\cdots+2f(x_{n-1})+f(x_n)]$$where $n$ is the number of subintervals, $a$ and $b$ are the lower and upper limits of integration, respectively, and $x_i=a+ih$ for $i=0,1,2,\cdots,n$, where $h=\frac{b-a}{n}$.In this problem, $a=0$, $b=1$, $n=7$, and $f(x)=\frac{1}{2x+4}$. So, $h=\frac{1-0}{7}=\frac{1}{7}$ and$$x_i=0+i\left(\frac{1}{7}\right)=\frac{i}{7} \quad \text{ for } i=0,1,2,\cdots,7.$$Therefore,$$\begin{split} T_7 &=\frac{1-0}{2(7)}\left[f(0)+2f\left(\frac{1}{7}\right)+2f\left(\frac{2}{7}\right)+\cdots+2f\left(\frac{6}{7}\right)+f(1)\right] \\ &=\frac{1}{14}\left[\frac{1}{4}+2\left(\frac{1}{6}\right)+2\left(\frac{1}{8}\right)+2\left(\frac{1}{10}\right)+2\left(\frac{1}{12}\right)+2\left(\frac{1}{14}\right)+\frac{1}{16}\right] \\ &=\frac{1}{14}\left(\frac{1}{4}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right) \\ &=\frac{1}{14}\left(\frac{7}{4}\right) \\ &=\frac{1}{8} \\ & \approx 0.1250. \end{split}$$Therefore, the correct choice is A.

Use the Trapezoidal Rule to Approximate the Integral with Answers ∫01dx2x+4;n=7\int _ { 0 } ^ { 1 } \frac { d x } { 2 x + 4 } ; \quad n = 7∫01​2x+4dx​;n=7

Use the Trapezoidal Rule to Approximate the Integral with Answers $\int _ { 0 } ^ { 1 } \frac { d x } { 2 x + 4 } ; \quad n = 7$