"Parmetric Arc Length "=int_a^b (sqrt((dx/dt)^2+((dy)/dt)^2))dtParmetric Arc Length =∫ba⎛⎝√(dxdt)2+(dydt)2⎞⎠dt
Differentiate using the chain rule:
dx/dt=1/2(t+2)^(-1/2)=frac{1}{2sqrt(t+2)}dxdt=12(t+2)−12=12√t+2
(dy)/dt=-1/t^2dydt=−1t2
"Arc Length " = int_1^3(sqrt((frac{1}{2sqrt(t+2)})^2+(-1/t^2)^2))dtArc Length =∫31(√(12√t+2)2+(−1t2)2)dt
Simplify a bit:
=int_1^3 (sqrt(frac{1}{4(t+2)}+1/t^4))dt=∫31(√14(t+2)+1t4)dt
=int_1^3 (sqrt(frac{t^4+4t+8}{4t^4(t+2) }))dt=∫31(√t4+4t+84t4(t+2))dt
Plug this into a calculator to get
"Arc Length " approx .863746Arc Length ≈.863746
