Length of an arc

### 12.3.8 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 24 and 25).
- arcLen(curve) returns the length of the curve.

#### Examples

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)

2 | √ |
5
| −ln | ⎛ ⎜ ⎝ | √ |
5
| −2 | ⎞ ⎟ ⎠
4

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

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

3
4

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))

1
2
| π

Compute the length of an arc:

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

1
4
| π | √ |
2