Arc Length

Core Sentence / 核心句

Arc length accumulates tiny straight-line approximations obtained from the Pythagorean theorem.

$$dL=\sqrt{(dx{)}^2+(dy{)}^2}.$$ $$L={\int }_a^b\sqrt{1+[f^{\prime }(x){]}^2}dx$$

and parametrically

$$L={\int }_a^b\sqrt{[x^{\prime }(t){]}^2+[y^{\prime }(t){]}^2}dt.$$

Connection / 连接

Parametric arc length is also distance traveled, because $\sqrt{[x^{\prime }{]}^2+[y^{\prime }{]}^2}$ is speed.

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