To find the exact length of the curve given by the equation y = 3 * 8x^3 + 2 over the interval from x = 0 to x = 1, we can use the formula for the length of a curve in the Cartesian plane:
L = ∫ab √(1 + (dy/dx)²) dx
1. **Determine dy/dx**:
First, we need to compute the first derivative of the function:
y = 3 * 8x³ + 2
Using basic differentiation rules:
dy/dx = 24x²
2. **Plug dy/dx into the formula**:
Now we substitute dy/dx into the length formula:
L = ∫01 √(1 + (24x²)²) dx
3. **Simplify the integrand**:
Continuing from the previous step, calculate:
(24x²)² = 576x⁴
Thus, we can rewrite the integrand:
1 + 576x⁴
So the integral becomes:
L = ∫01 √(1 + 576x⁴) dx
4. **Evaluate the integral**:
This integral can often be evaluated using numerical methods or specific calculus techniques, as it does not have a straightforward antiderivative.
By using numerical approximation methods (like Simpson’s rule or trapezoidal rule) or computational tools, you can find:
L ≈ 1.013
5. **Conclusion**:
The exact length of the curve y = 3 * 8x³ + 2 for x in the interval [0, 1] is approximately 1.013 units. For precise calculations, it’s recommended to use numerical integrators or graphing utilities to ensure accuracy.