Mathematics

Mathematics

Problem

For the definite integral ∫25(x2−1) dx\displaystyle \int_{2}^{5} (x^{2}-1)\,dx, the interval [2,5][2,5] is divided into nn equal subintervals of width Δx\Delta x, and the midpoint of each subinterval is chosen as the sample point. Which of the following limits represents the integral? a) lim⁡n→∞∑i=1n[(2+3in)2−1]3n\displaystyle \lim_{n\to\infty}\sum_{i=1}^{n}\left[\left(2+\frac{3i}{n}\right)^{2}-1\right]\frac{3}{n} b) lim⁡n→∞∑i=1n[(2+3(i−12)n)2−1]3n\displaystyle \lim_{n\to\infty}\sum_{i=1}^{n}\left[\left(2+\frac{3(i-\tfrac12)}{n}\right)^{2}-1\right]\frac{3}{n} c) lim⁡n→∞∑i=1n(2+3(i−12)n)23n\displaystyle \lim_{n\to\infty}\sum_{i=1}^{n}\left(2+\frac{3(i-\tfrac12)}{n}\right)^{2}\frac{3}{n} d) lim⁡n→∞∑i=1n[(2+3(i−12)n)2−1]1n\displaystyle \lim_{n\to\infty}\sum_{i=1}^{n}\left[\left(2+\frac{3(i-\tfrac12)}{n}\right)^{2}-1\right]\frac{1}{n}

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