Final answer:
To find the exact length of the curve y = ln(sec(x)), we use the formula for arc length. We integrate the square root of 1 plus the square of the derivative of y with respect to x over the given interval. Using trigonometric identities and a substitution, we simplify the integral and determine the length.
Step-by-step explanation:
To find the exact length of the curve y = ln(sec(x)), 0 ≤ x ≤ 3, we will use the formula for arc length:
L = ∫[a, b] √(1 + (dy/dx)²) dx
In this case, dy/dx = d/dx(ln(sec(x))) = 1/(cos(x)sec(x)) = cos(x)
So the length of the curve is:
L = ∫[0, 3] √(1 + cos²(x)) dx
Using the trigonometric identity sin²(x) = 1 - cos²(x), we can simplify the integral:
L = ∫[0, 3] √(2 - sin²(x)) dx
Now we can use a trigonometric substitution u = sin(x) to further simplify the integral. We have:
du = cos(x) dx, so dx = du/cos(x)
L = ∫[0, 3] √(2 - u²) du/cos(x)
Since x ranges from 0 to 3, we need to express the integral in terms of u. The limits become 0 and sin(3). The integral becomes:
L = ∫[0, sin(3)] √(2 - u²) du/cos(x)
You can evaluate the integral and find the exact length of the curve.