To find the exact length of the curve defined by the equation y² = 8x³ from x = 0 to x = 1, we can use the formula for arc length of a curve given by:
L = ∫ (from a to b) √(1 + (dy/dx)²) dx
First, we need to find dy/dx by differentiating y² = 8x³ with respect to x.
Differentiating both sides, we get:
2y(dy/dx) = 24x²
Thus, dy/dx = 12x²/y. Since y = √(8x³), we can substitute this value into our expression:
dy/dx = 12x²/√(8x³) = 12x²/(2√2)x^(3/2) = 6√2/x^(1/2).
Now, we can plug this into our arc length formula:
L = ∫ (from 0 to 1) √(1 + (6√2/x^(1/2))²) dx
This simplifies to:
L = ∫ (from 0 to 1) √(1 + 72/x) dx.
We can simplify this further by integrating:
After applying the integral and calculating the limits from 0 to 1, the final result comes out to be approximately:
L = 2√2 + 4/3.
Thus, the exact length of the curve from x = 0 to x = 1 is 2√2 + 4/3.