How can I find the exact length of the curve defined by the equation y = 18x^3 + 20x + 1?

by

To find the exact length of the curve defined by the equation y = 18x3 + 20x + 1, we use the formula for the arc length of a curve. The arc length L of a function y = f(x) from x = a to x = b is given by:

L = ∫ab √(1 + (dy/dx)2) dx

Here’s a step-by-step process to find the length of the curve:

  1. Find dy/dx: Compute the derivative of the function.

    2. The derivative of y with respect to x is:

    dy/dx = 54x2 + 20

  2. Calculate (dy/dx)2: Square the derivative.

    3. (dy/dx)2 = (54x2 + 20)2

  3. Set up the integral: Substitute into the arc length formula.

    4. The expression under the integral becomes:

    √(1 + (54x2 + 20)2)

  4. Select the limits a and b: Define the interval over which you want to calculate the length of the curve. For example, let’s say we want to find the length from x = 0 to x = 1.
  5. Calculate the integral: Perform the definite integral using numerical methods or calculus techniques.

    6. After evaluating the integral, you will arrive at the precise length of the curve within the specified interval.

This generalized approach allows you to compute the curve length for any defined limits of x. You can also use numerical integration tools for more complicated expressions or vast ranges of x.

Leave a Comment