Solve the problem.
-Estimate the minimum number of subintervals needed to approximate the integral $\int _ { 3 } ^ { 6 } \frac { 1 } { ( x - 1 ) ^ { 2 } } d x$ with an error of magnitude less than $10 ^ { - 4 }$ using the Trapezoidal Rule.
A) 368
B) 15
C) 46
D) 92

Question

Quizplus · Accepted Answer

The error bound for the Trapezoidal Rule is given by $\frac{(b-a)^3}{12n^2} \max |f''(x)|$ on [a, b]. Here, $f(x) = \frac{1}{(x-1)^2}$, so $f''(x) = \frac{6}{(x-1)^4}$. The maximum value of $f''(x)$ on [3, 6] occurs at x = 3, which is $\frac{6}{(3-1)^4} = \frac{6}{16} = \frac{3}{8}$. Setting the error less than $10^{-4}$ and solving for n gives $n \approx 92$.

Solve the Problem ∫361(x−1)2dx\int _ { 3 } ^ { 6 } \frac { 1 } { ( x - 1 ) ^ { 2 } } d x∫36​(x−1)21​dx

Solve the Problem $\int _ { 3 } ^ { 6 } \frac { 1 } { ( x - 1 ) ^ { 2 } } d x$