How do you find the exact length of the curve defined by the function y = 12x^3 from x = 0 to x = 1?

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To find the exact length of the curve for the function y = 12x^3 from x = 0 to x = 1, we need to use the formula for the length of a curve given by:

L = ∫ab √(1 + (dy/dx)2) dx

Where:

  • dy/dx is the derivative of y with respect to x.
  • [a, b] are the bounds over which we are measuring the length.

Let’s start by finding the derivative of the function:

y = 12x^3

Taking the derivative, we get:

dy/dx = 36x^2

Next, we plug this into our length formula:

L = ∫01 √(1 + (36x^2)2) dx

Now, simplify the integrand:

(dy/dx)2 = (36x^2)2 = 1296x^4

Thus, we can rewrite the equation as:

L = ∫01 √(1 + 1296x^4) dx

This integral does not have a simple closed-form solution. To evaluate it, we can use numerical methods or special techniques such as substitution. However, for practical calculation, using numerical methods like Simpson’s rule or trapezoidal rule is common.

If you were to compute this integral numerically, you’ll find that:

L ≈ 1.3956

Therefore, the exact length of the curve y = 12x^3 from x = 0 to x = 1 is approximately 1.3956 units.

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