To find the exact length of the curve defined by the function y = ln(1 + x²) over the interval [0, 1.5], we can use the formula for the arc length of a curve in Cartesian coordinates:
L = ∫ab √(1 + (dy/dx)²) dx
Here, dy/dx is the derivative of the function with respect to x. First, we need to compute the derivative:
y = ln(1 + x²)
Then, using the chain rule, we find:
dy/dx = (1 / (1 + x²)) * (2x) = (2x) / (1 + x²)
Next, we square the derivative:
(dy/dx)² = [(2x) / (1 + x²)]² = (4x²) / (1 + x²)²
Now we substitute this into the arc length formula:
L = ∫01.5 √(1 + (4x²) / (1 + x²)²) dx
We simplify the expression under the square root:
1 + (4x²) / (1 + x²)² = (1 + x²)² + 4x² = (1 + 2x² + x⁴) + 4x² = x⁴ + 6x² + 1
The integral now becomes:
L = ∫01.5 √(x⁴ + 6x² + 1) dx
This integral can be computed, usually requiring techniques such as substitution or numerical approximation, as it does not have a simple antiderivative. However, for an exact length, it can also be evaluated using numerical methods or definite integral calculators.
The final value of the integral provides the exact length of the curve for the given interval. Depending on the method used for computing the integral, the approximate length can be found.
Hence, the exact length of the curve y = ln(1 + x²) from x = 0 to x = 1.5 is determined through careful calculation or numerical integration.