Length of Arc in XY-Plane | Applications of Integration

 

 

Apply Pythagorean theorem to the triangular strip shown in the figure:
(ds)2=(dx)2+(dy)2   ←   Equation (1)
 

length-of-arc-xy-plane.jpg

 

Divide both sides of Equation (1) by (dx)2
(ds)2(dx)2=(dx)2(dx)2+(dy)2(dx)2

(dsdx)2=1+(dydx)2

dsdx=1+(dydx)2

ds=1+(dydx)2dx
 

Integrate both sides

s=x2x11+(dydx)2dx

 

Divide both sides of Equation (1) by (dy)2
(ds)2(dy)2=(dx)2(dy)2+(dy)2(dy)2

(dsdy)2=(dxdy)2+1

dsdy=(dxdy)2+1

ds=(dxdy)2+1 dy

 

Integrate both sides

s=y2y1(dxdy)2+1 dy