Trapezoidal method integration and error estimation

Determine quantity of subintervals you need to split integration interval to calculate: $$\int_{0}^{1} (x^2+1)\, dx$$ with error less or equal to $0.01$ using the composite trapezoidal rule. Calculate the value of this integral after splitting the interval into $3$ parts. Estimate an error.

First I calculated integral with splitting it in three parts and got answer $1.3266$ which is near with actual $1.33$ result. Then I tried this formula to calculate quantity of subintervals: $$|\delta|\le \max|f''(x)|\,\frac{(b-a)^3}{12n^2}$$ and I got that $|n|=4.08$. Using the same formula I calculated that for $3$ intervals estimate error is $$|\delta| \le 0.018$$

Question: is this calculations and formulas usage are right ?

• It's not easy to split an interval into $4.08$ subintervals. You might want to make that $5$. – Robert Israel May 18 '17 at 21:14

1 Answer

Since $f(x) = x^2+1$, then $f''(x) = 2$. Also $a=0, b=1$, then $$|\delta| \leq \frac{\max|f''(x)|(b-a)^3}{12n^2} = \frac{2}{12n^2} = \frac{1}{6n^2}.$$

With an error less than $0.01$ we have $\frac{1}{6n^2} \leq 0.01$, so $n \geq \sqrt{\frac{100}{6}} = 4.08$. So have to take $n = 5$ to acheive the required error.

For three subintervales $$|\delta| \leq \frac{1}{54} \simeq 0.019.$$