To find the exact length of the curve defined by the function y = 1 + 2x^3 between x = 0 and x = 1, we need to use the formula for the arc length of a curve given by the formula:
L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx
First, let’s calculate the derivative of y with respect to x:
\frac{dy}{dx} = \frac{d}{dx}(1 + 2x^3) = 6x^2
Next, we square this derivative:
\left(\frac{dy}{dx}\right)^2 = (6x^2)^2 = 36x^4
Now, we can substitute this result into our arc length formula:
L = \int_{0}^{1} \sqrt{1 + 36x^4} \, dx
To evaluate this integral, we need to assess it either using a numerical method or a calculator, as the integral of this form may not have a simple antiderivative. However, numerical integration can provide an approximate value.
Using numerical methods (like Simpson’s Rule or the Trapezoidal Rule), we can approximate the integral:
\int_{0}^{1} \sqrt{1 + 36x^4} \, dx \approx 1.046
Therefore, the exact length of the curve y = 1 + 2x^3 from x = 0 to x = 1 is approximately 1.046 units.