Length of an arc

The arcLen command finds the lengths of curves in the plane, which can either be given by an equation or a curve object.

To find the length of a curve given by an equation, arcLen takes four arguments:
- expr, an expression (resp. a list of two expressions [expr₁,expr₂]) involving a variable x.
- x, the name of the variable.
- a and b, two values for the bounds of this variable.
arcLen(expr,x,a,b) resp. arcLen([expr₁,expr₂]x,a,b) returns the length of the curve defined by y=f(x)=expr resp. by x₁=expr₁, x₂=expr₂ as x varies from a to b, using the formula
arcLen(f(x),x,a,b)= ∫
b

a

√

f′(x)²+1

dx

or
arcLen(f(x),x,a,b)= ∫
b

a

√

x′(t)²+y′(t)²

dt.
To find the length of a curve given by a curve object, arcLen takes a single argument: curve, a geometric curve defined in one of the graphics chapters (chapters 26 and 27).
arcLen(curve) returns the length of the curve.

Compute the length of the parabola y=x² from x=0 to x=1:

arcLen(x^2,x,0,1)

or:

arcLen([t,t^2],t,0,1)

√

−ln

⎛
⎜
⎝

√

−2

⎞
⎟
⎠

Compute the length of the curve y=cosh(x) from x=0 to x=ln(2):

arcLen(cosh(x),x,0,log(2))

Compute the length of the circle x=cos(t),y=sin(t) from t=0 to t=2π:

arcLen([cos(t),sin(t)],t,0,2*pi)

2 π

Compute the length of the unit circle segment in the first quadrant:

arcLen(circle(0,1,0,pi/2))

Compute the length of an arc:

arcLen(arc(0,1,pi/2))

√