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)
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=∫x2x1√1+(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