【微分積分】4-3-2 区分求積法｜問題集

\(\displaystyle =\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^{n}\frac{k}{n}\)
\(\displaystyle =\int_0^1xdx\)
\(\displaystyle =\left[\frac{x^2}{2}\right]_0^1\)
\(\displaystyle =\frac{1}{2}\)

\(\displaystyle =\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^{n}\frac{1}{2+\frac{k}{n}}\)
\(\displaystyle =\int_0^1\frac{1}{2+x}dx\)
\(\displaystyle =[\log|2+x|]_0^1\)
\(\displaystyle =\log3-\log2\)
\(\displaystyle =\log\frac{3}{2}\)

\(\displaystyle =\int_0^1\sqrt{x}dx\)
\(\displaystyle =\left[\frac{2}{3}x^{\frac{3}{2}}\right]_0^1\)
\(\displaystyle =\frac{2}{3}\)

\(\displaystyle =\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^{n}\frac{1}{1+\frac{k}{n}}\)
\(\displaystyle =\int_1^2\frac{1}{x}dx\)
\(\displaystyle =[\log|x|]_1^2\)
\(\displaystyle =\log2-\log1\)
\(\displaystyle =\log2\)

\(\displaystyle =\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^{n}\sqrt{\frac{1}{1+(\frac{k}{n})^2}}\)
\(\displaystyle =\int_0^1\sqrt{\frac{1}{1+x^2}}dx\)
\(\displaystyle =[\log|x+\sqrt{1+x^2}|]_0^1\)
\(\displaystyle =\log(1+\sqrt{2})\)