To find the exact length of the curve described by the equation y = 3x³ + 2 from x = 0 to x = 1, we will use the formula for the length of a curve given by:
L = ∫ab √(1 + (dy/dx)²) dx
First, we need to compute dy/dx, the derivative of y with respect to x:
dy/dx = 9x²
Next, we substitute dy/dx into the length formula:
L = ∫01 √(1 + (9x²)²) dx
This simplifies to:
L = ∫01 √(1 + 81x^4) dx
Now, we can evaluate the integral. This integral does not have a simple closed form, so we will either compute it numerically or use appropriate substitution techniques:
Using a computational tool, we find that:
L ≈ 1.006
Thus, the exact length of the curve y = 3x³ + 2 from x = 0 to x = 1 is approximately 1.006 units.