Solve the problem.
-Estimate the minimum number of subintervals needed to approximate the integral $\int _ { 1 } ^ { 4 } \left( 4 x ^ { 2 } + 7 \right) d x$ with an error of magnitude less than $10 ^ { - 4 }$ using the Trapezoidal Rule.
A) 213
B) 849
C) 425
D) 82

Question

Quizplus · Accepted Answer

Using the formula for the error in the Trapezoidal Rule:

$$E_T\leq\frac{b-a}{12}h^2\max_{a\leq x\leq b}|f''(x)|$$

We can estimate the minimum number of subintervals needed as follows:

$$\frac{b-a}{12}h^2\max_{a\leq x\leq b}|f''(x)|\leq 10^{-4}$$
$$h=\frac{b-a}{n}=\frac{3}{n}$$
$$4\left(\frac{3}{n}\right)^2\max_{1\leq x\leq 4}|8|=\frac{36}{n^2}\cdot 8 \leq 10^{-4}$$
$$n\geq\sqrt{8\cdot 10^4\cdot 9}=424.26 \approx 425$$

Therefore, we need at least 425 subintervals. The closest option is C) 425.

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