Use the Trapezoidal Rule with n = 4 steps to estimate the integral.
-$$\int _ { 1 } ^ { 3 } \frac { 1 } { x ^ { 2 } } d x$$
A) $\frac { 141 } { 200 }$
B) $\frac { 141 } { 400 }$
C) $\frac { 71 } { 100 }$
D) $\frac { 141 } { 100 }$

Question

Quizplus · Accepted Answer

The Trapezoidal Rule formula for estimating an integral $\int_a^b f(x) dx$ with $n$ steps is $\frac{b-a}{2n}[f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n)]$, where $x_0 = a$, $x_n = b$, and $x_i = a + i\frac{b-a}{n}$ for $i = 1, 2, \ldots, n-1$. For $\int _ { 1 } ^ { 3 } \frac { 1 } { x ^ { 2 } } d x$ with $n = 4$, the interval $[1, 3]$ is divided into 4 steps: $x_0 = 1$, $x_1 = 1.5$, $x_2 = 2$, $x_3 = 2.5$, and $x_4 = 3$. Applying the formula:$\frac{3-1}{2*4}[f(1) + 2f(1.5) + 2f(2) + 2f(2.5) + f(3)] = \frac{1}{4}[1 + 2(\frac{4}{9}) + 2(\frac{1}{4}) + 2(\frac{4}{25}) + \frac{1}{9}]$ = $\frac{1}{4}[\frac{141}{100}]$ = $\frac{141}{400}$.However, there seems to be a mistake in my calculation. Correcting it:$\frac{3-1}{8}[1 + 2(\frac{4}{9}) + 2(\frac{1}{4}) + 2(\frac{4}{25}) + \frac{1}{9}]$ = $\frac{1}{4}[\frac{141}{50}]$ = $\frac{141}{200}$.Thus, the correct answer is $\frac{141}{200}$, which matches option A.

Use the Trapezoidal Rule with N = 4 Steps to Estimate