Solve the problem.
-Estimate the minimum number of subintervals needed to approximate the integral $\int _ { 1 } ^ { 3 } \left( 6 x ^ { 4 } - 5 x \right) d x$ with an error of magnitude less than $10 ^ { - 4 }$ using Simpson's Rule.
A) 38
B) 30
C) 24
D) 84

Question

Quizplus · Accepted Answer

Using Simpson's Rule, the error can be estimated by: $$E_n \approx \frac{(b-a)^5}{180n^4}f^{(4)}(\xi)$$ where $n$ is the number of subintervals, $a$ and $b$ are the limits of integration, and $\xi$ is some point in the interval $[a,b]$. Here, $f^{(4)}(x)=480x^2$ is the fourth derivative of $(6x^4-5x)$. We want $E_n<10^{-4}$. Substituting all the values, we get: $$\frac{(3-1)^5}{180n^4}(480 \cdot \xi^2) < 10 ^ { - 4 }$$ $$\Rightarrow \xi^2<\frac{5n^4}{32}$$ The maximum value of $\xi^2$ is at $x=3$, so we use that: $$9<\frac{5n^4}{32}$$ $$\Rightarrow n^4>\frac{288}{5}$$ $$\Rightarrow n>4.34$$ So we need a minimum of 5 subintervals. Checking the answer choices, only option C has more than 5 subintervals, so the answer is C.

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