arclength of a curve problem?

Question

The length of a curve from x=1 to x=4 is given by int(from 1 to 4) of sqrt(1+9x^4) dx. If the curve contains the point (1,6), which of the following could be an equation for this curve?

y=3+3x^2
y=5+x^3
y=6+x^3
y=6-x^3
y=(16/5)+x+(9/5)x^5

Answer 1

y = 5 + x^3

Since you're integrating the differential length of the arc, it's of the form √((dx/dx)² + (dy/dx)²) so that the dy/dx term must be √(9x^4) = 3x². The only only equation containing the integral of 3x² (namely, x^3) containing the point (1,6) is the 2'nd one.

HTH

Doug



Doug