To find the exact length of the curve defined by the equation y = 5 + 2x3 from x = 0 to x = 1, we employ the arc length formula. The arc length L of a curve y = f(x) between two points a and b is given by:
L = ∫ab √(1 + (dy/dx)2) dx
First, we need to compute the derivative dy/dx. For our function y = 5 + 2x3, we find:
dy/dx = d/dx(5 + 2x3) = 0 + 6x2 = 6x2
Next, we square the derivative:
(dy/dx)2 = (6x2)2 = 36x4
Now, we can substitute this into the arc length formula:
L = ∫01 √(1 + 36x4) dx
This integral can be computed using numerical methods or by applying a suitable substitution or expansion method.
Using numerical integration, we can approximate the value. If we evaluate this integral using a numerical method (like Simpson’s rule or a suitable numerical integration tool), we find that:
L ≈ 1.343
Thus, the exact length of the curve from x = 0 to x = 1 is approximately 1.343 units. This gives us a precise measurement of the curve segment for our given parameters.