Which of the following?

Question

The length of a curve from x=0 to x=1 is given by ∫ sqrt(1+16x^6) dx from 0 to 1. If the curve contains the point (1, 4), which of the following could be an equation for this curve? 

 

a) y=x^4+3

b) y=x^4+1

c) y=1+16x^6

d) y=sqrt(1+16x^6)

e) y=x+x^7/7

 

Answer: A.

 

Please show all your work.

Answer 1

We have the equation for length given by L = ∫(dx/dx^2 + dy/dx^2)^(1/2) dx.  This tells us that dy/dx ^2 = 16 x^6.  Thus, dy/dx = 4x^3.  Integrating with respect to x gives us y = x^4 +c.  Substituting in the point (1,4) gives 4 = 1 + c.  So c =3.  Thus, y = x^4 + 3.