SciTech-Mathmatics-Analysis:
定積分 求解的“十大公式”
1. Newton-Leibniz formula
\(\large \begin{array}{rl} \\ \int_{a}^{b}{f'(x) dx} =& f(b) - f(a) \\ =& \underset{n \rightarrow \infty}{\lim} \overset{ n }{\underset{k=1}{\sum}} { ( f'(x_k) \cdot \Delta{x_k} ) }, 黎曼和形式\\ =& \underset{n \rightarrow \infty}{\lim} \overset{ n }{\underset{k=1}{\sum}} { \Delta{f(x_k)} },\ 無窮微分增量形 \\ \end{array}\)
2. the geometry representation of Definition of The Definite Integral
for example,
- \(\large \int_{0}^{a}{\sqrt{a^2 - x^2} dx} =\frac{\pi}{4} a^2\),
for \(\large y =\sqrt{a^2 - x^2}\) <=> \(\large x^2 + y^2 = a^2\),
a quarter of \(\large a\ circle\) to which it's radius equal to \(\large a\) - \(\large \int_{0}^{a}{\sqrt{2 a x - x^2} dx} = \frac{\pi}{4} a^2\),
for \(\large y =\sqrt{-(x-a)^2 + a^2}\) <=> \(\large (x-a)^2 + y^2 = a^2\),
a quarter of \(\large a\ circle\) to which it's radius equal to \(\large a\) also.