To find the exact length of the curve defined by the function y = ln(1 + x^2) from x = 0 to x = 18, we use the arc length formula. The formula for the length of a curve given by y = f(x) from a to b is:
L = ∫_a^b √(1 + (dy/dx)^2) dxFirst, we need to find dy/dx. Differentiating y with respect to x, we get:
dy/dx = (2x) / (1 + x^2)Now we can compute ((dy/dx)^2):
((dy/dx)^2) = ((2x)^2) / (1 + x^2)^2 = (4x^2) / (1 + x^2)^2Next, we substitute this back into the arc length formula:
L = ∫_0^18 √(1 + (4x^2) / (1 + x^2)^2) dxWe simplify the expression:
L = ∫_0^18 √((1 + x^2)^2 + 4x^2) / (1 + x^2) dxThis simplifies further to:
L = ∫_0^18 (1 + x^2) / (1 + x^2) dx = ∫_0^18 dxThus, the actual length integral simplifies to:
L = [x]_0^18 = 18 - 0 = 18Therefore, the exact length of the curve from x = 0 to x = 18 is 18 units.