Solve the problem.
-Estimate the minimum number of subintervals needed to approximate the integral $\int _ { - \pi / 2 } ^ { \pi / 2 } 4 \sin x d x$ with an error of magnitude less than $10 ^ { - 4 }$ using the Trapezoidal Rule.
A) 58
B) 114
C) 322
D) 228

Question

Quizplus · Accepted Answer

The error in the Trapezoidal Rule is given by $|E_T| \leq \frac{(b-a)^3}{12n^2} \max |f''(x)|$. For $f(x) = 4\sin x$, $f''(x) = -4\sin x$, and $\max |f''(x)| = 4$. The interval $[- \pi/2, \pi/2]$ has length $\pi$. Solving $\frac{\pi^3}{12n^2} \cdot 4 < 10^{-4}$ gives $n \approx 322$.

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