[spec] add formula for length of a parametric curve

diff --git a/spec/calculus.md b/spec/calculus.md
index 30a295788bca6e503648f0a6a247a7cbd3957882..43cf752fc9da76c5fece553dbecaeed0be5d433f 100644 (file)
--- a/spec/calculus.md
+++ b/spec/calculus.md
@@ -347,13 +347,11 @@ where $f(x) > g(x)$
 
 ## Length of a curve
 
 
 ## 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
 
 
 ## Rates
 


@@ -361,7 +359,7 @@ $$l = \int^b_a \sqrt{({dx \over dt})^2 + ({dy \over dt})^2} \> dt$$
 
 $${da \over db} \quad \text{(change in } a \text{ with respect to } b)$$
 
 
 $${da \over db} \quad \text{(change in } a \text{ with respect to } b)$$
 

-#### Gradient at a point on parametric curve
+### Gradient at a point on parametric curve

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