Solve the problem.
-Estimate the minimum number of subintervals needed to approximate the integral $\int _ { 0 } ^ { 5 } \sqrt { x + 4 } d x$ with an error of magnitude less than $10 ^ { - 4 }$ using Simpson's Rule.
A) 7
B) 8
C) 4
D) 6

Question

Quizplus · Accepted Answer

Let $f(x)=\sqrt{x+4}.$ We have $f''(x)=\frac{-1}{4\sqrt{(x+4)^3}}.$ Since $f''(x)\le0$ for $x\ge0,$ we have $\max_{0\le x\le5}|f''(x)|=|f''(0)|=\frac{-1}{16}.$ Thus, we have the error bound
$$\frac{(5-0)^5}{180n^4}\cdot\max_{0\le x\le5}|f''(x)|=\frac{125}{18n^4}.$$
We want to find $n$ such that the error bound is less than $10^{-4},$ or
$$\frac{125}{18n^4}<10^{-4}.$$
Solving for $n,$ we have
$$n>\left(\frac{625\cdot18}{1\cdot10^4}\right)^{\frac{1}{4}}\approx5.76$$
Since $n$ must be an integer, $n\ge6.$ Trying $n=6,$ we find that Simpson's Rule with $n=6$ subintervals gives
$$\left|\int_0^5f(x)\,dx-\frac{5}{45}\left(f(0)+4f(2.5)+f(5)\right)\right|\approx7\cdot10^{-5}.$$
Thus, the minimum number of subintervals needed is $\boxed{\textbf{(D) }6}.$

Solve the Problem ∫05x+4dx\int _ { 0 } ^ { 5 } \sqrt { x + 4 } d x∫05​x+4​dx

Solve the Problem $\int _ { 0 } ^ { 5 } \sqrt { x + 4 } d x$