From e94ea5cc8fa3c59c54a0607de164e0caa0aec081 Mon Sep 17 00:00:00 2001 From: Andrew Lorimer Date: Sun, 19 May 2019 14:15:04 +1000 Subject: [PATCH] [spec] add formula for length of a parametric curve --- spec/calculus.md | 10 ++++------ 1 file changed, 4 insertions(+), 6 deletions(-) diff --git a/spec/calculus.md b/spec/calculus.md index 30a2957..43cf752 100644 --- a/spec/calculus.md +++ b/spec/calculus.md @@ -347,13 +347,11 @@ where $f(x) > g(x)$ ## Length of a curve -$$L = \int^b_a \sqrt{1 + ({dy \over dx})^2} \> dx$$ +$$L = \int^b_a \sqrt{1 + ({dy \over dx})^2} \> dx \quad \text{(Cartesian)}$$ -Evaluate on CAS. Or use Interactive $\rightarrow$ Calculation $\rightarrow$ Line $\rightarrow$ `arcLen`. - -### Parametric curve +$$L = \int^b_a \sqrt{{dx \over dt} + ({dy \over dt})^2} \> dt \quad \text{(parametric)}$$ -$$l = \int^b_a \sqrt{({dx \over dt})^2 + ({dy \over dt})^2} \> dt$$ +Evaluate on CAS. Or use Interactive $\rightarrow$ Calculation $\rightarrow$ Line $\rightarrow$ `arcLen`. ## Rates @@ -361,7 +359,7 @@ $$l = \int^b_a \sqrt{({dx \over dt})^2 + ({dy \over dt})^2} \> dt$$ $${da \over db} \quad \text{(change in } a \text{ with respect to } b)$$ -#### Gradient at a point on parametric curve +### Gradient at a point on parametric curve $${dy \over dx} = {{dy \over dt} \div {dx \over dt}} \> \vert \> {dx \over dt} \ne 0$$ -- 2.43.2