Final answer:
The length of the curve y = 8 cos x from 0 to 2 is given by the integral S = integral from 0 to 2 of the square root of (1 + 64 sin^2 x) dx, which is set up using the formula for arc length in Cartesian coordinates.
Step-by-step explanation:
To set up an integral for the length of the curve y = 8 cos x, with the range 0 ≤ x ≤ 2, we use the formula for the arc length of a curve in Cartesian coordinates:
The arc length S can be found using the formula:
S = ∫ √(1 + (dy/dx)^2) dx
Here, y = 8 cos x is our function. To find dy/dx, we differentiate y with respect to x to get:
dy/dx = -8 sin x
Now, we substitute dy/dx into the arc length formula:
S = ∫ √(1 + (-8 sin x)^2) dx
The given range for x is from 0 to 2, so the integral will be:
S = ∫_{0}^{2} √(1 + 64 sin^2 x) dx
This integral is set up but not evaluated. To find the actual length, the integral would need to be calculated, potentially using numerical methods.