Arclength problems?

Question

find the arc length of the graph of: y=1/3((x^2)+2)^(3/2)

Here is the limit x [0,2]

Answer 1

to find arclength, integrate sqrt(1 + (dy/dx)^2). in this case,
dy/dx = x*(x^2 + 2)^(1/2), so the integrand is

sqrt(1 + x^2*(x^2+2)) = sqrt(1 + 2x^2 + x^4).

now we're in luck, because the thing in the square root is a perfect square:
sqrt(1 + 2x^2 + x^4) = sqrt((1 + x^2)^2)
= 1 + x^2.

so the arclength is int[0 to 2] (1+x^2)dx, which I'm assuming you can do.

Answer 2

14/3