Solve the problem.
-Estimate the minimum number of subintervals needed to approximate the integral $\int _ { 5 } ^ { 6 } \left( 4 x ^ { 3 } + 1 x \right) d x$ with an error of magnitude less than $10 ^ { - 4 }$ using the Trapezoidal Rule.
A) 142
B) 317
C) 347
D) 174

Question

Quizplus · Accepted Answer

Using the Trapezoidal Rule, the error can be bounded by the formula:

$$E_n\leq \frac{(b-a)^3}{12n^2}M$$

where $n$ is the number of subintervals, $M=\max_{a\leq x\leq b} |f''(x)|$ and $a=5$, $b=6$.

Taking the second derivative of $f(x)=4x^3+x$ yields $f''(x)=24x$. To find $M$, we need to find the maximum value of $f''(x)$ on the interval $[5,6]$. This value occurs at $x=6$ and is $M=144$.

Plugging in $a=5$, $b=6$, $M=144$, and $E_n=10^{-4}$, we have:

$$10^{-4}\leq \frac{(6-5)^3}{12n^2}(144)$$

Simplifying and solving for $n$, we get:

$$n\geq \sqrt{\frac{(6-5)^3}{12(10^{-4})}(144)}\approx 347.42$$

Since $n$ must be a positive integer, the minimum number of subintervals needed is $\boxed{	extbf{(C) }347}$.

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