From: Andrew Lorimer <andrew@lorimer.id.au>
Date: Sun, 19 May 2019 04:15:04 +0000 (+1000)
Subject: [spec] add formula for length of a parametric curve
X-Git-Tag: yr12~131
X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/e94ea5cc8fa3c59c54a0607de164e0caa0aec081?ds=sidebyside

[spec] add formula for length of a parametric curve
---

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
 
 $${dy \over dx} = {{dy \over dt} \div {dx \over dt}} \> \vert \> {dx \over dt} \ne 0$$