Use the error formula to estimate the error in approximating the integral $\int _ { 0 } ^ { \pi } 3 \cos x d x$ with $n = 6$ using Simpson's Rule. Round your answer to six decimal places.
A) $0.007084$
B) $0.141676$
C) $0.004723$
D) $0.003936$
E) $0.002530$

Question

Quizplus · Accepted Answer

The error in approximating the integral using Simpson's Rule is given by the formula:$$E_n \leq \frac{K_4(b-a)^5}{180n^4}$$where $K_4$ is the maximum value of the fourth derivative of the function on the interval $[a,b]$.In this case, the function is $f(x) = 3\cos x$, and its fourth derivative is $f^{(4)}(x) = -3\cos x$. The maximum value of $\cos x$ on the interval $[0,\pi]$ is 1, so $K_4 = 3$.Therefore, the error is:$$E_6 \leq \frac{3(\pi-0)^5}{180(6)^4} = 0.003936$$So the correct answer is D.

Use the Error Formula to Estimate the Error in Approximating ∫0π3cos⁡xdx\int _ { 0 } ^ { \pi } 3 \cos x d x∫0π​3cosxdx

Use the Error Formula to Estimate the Error in Approximating $\int _ { 0 } ^ { \pi } 3 \cos x d x$