Question

can someone help me with this problem dx/dt = sqrt(1+sin(t^2)). I only know how to separate variables sqrt(1+sin(t^2))dt = dx

Answer 1

The problem statement, all variables and given/known data
Find the area of the surface obtained by rotating the curve about the x-axis:
y=cos 2x, 0<=x<=pi/6

2. Relevant equations
Surface area about the x-axis = Integral of 2pi * f(x) * sqrt(1+[f'(x)]^2) dx

3. The attempt at a solution
I think I set up my integral correctly, so my problem is figuring out how to solve the integral. I have:

2pi * Integral of cos 2x * sqrt(1+4sin^2 2x) dx on [0, pi/6]

I can't figure out how to solve that. And can someone teach me how to use Latex for integrals so it looks better.

Answer 2

I haven't learned how to find surface about the x-axis. I'm just trying to differentiate

Answer 3

separate then differentiate

Answer 4

dx/dt = sqrt(1+sin(t^2))
dx= sqrt(1+sin(t^2)) dt

Answer 5

C(x) is fresnel C integral
and S(x) is fresnel S integral

Answer 6

is there any way to do it with the Fundamental Theorem?