Which of the following?

Question

Which of the following gives the length of the path described by the parametric equations x=(1/4)t^4 and y=t^3, where 0≤t≤2?

 

a) ∫ t^6+9t^4 dt from 0 to 2

 

b) ∫ sqrt(t^6+1) dt from 0 to 2

 

c) ∫ sqrt(1+9t^4) dt from 0 to 2

 

d) ∫ sqrt(t^6+9t^4) dt from 0 to 2

 

e) ∫ sqrt(t^3+3t^2) dt from 0 to 2

 

Please show all your work.

Answer 1

The equation for length is ∫((dx/dt ^2 + dy/dt ^2)^(1/2)) dt from 0 to 2.

So dx/dt = t^3 and dy/dt = 3t^2. 

So dx/dt ^2 = t^6 and dy/dt ^2 = 9t^4.

So our final solution is ∫ (t^6 + 9*t^4) ^(1/2) dt from 0 to 2 which is precisely answer d.