To find the exact length of the curve defined by the function y = x^3 – 3x + 4, we can use the arc length formula:
L = ∫ from a to b √(1 + (dy/dx)²) dx
First, we need to find the derivative (dy/dx):
dy/dx = 3x² – 3
Next, we need to calculate (dy/dx)²:
(dy/dx)² = (3x² – 3)² = 9x^4 – 18x² + 9
Now we can substitute this back into the arc length formula:
L = ∫ from a to b √(1 + (9x⁴ – 18x² + 9)) dx
After simplifying the expression inside the square root, we have:
L = ∫ from a to b √(9x⁴ – 18x² + 10) dx
At this point, we would evaluate the definite integral from a to b, where these limits depend on the specific interval for which you want to find the length. If you are given specific values for a and b, substitute them in and perform the integration, which may require numerical methods or further algebraic manipulation depending on the complexity.
In conclusion, calculating the exact length involves integration and the limits of integration (a and b) which are determined by the specific segment of the curve you wish to analyze.