How do you find the arc length function for the curve defined by the equation y = 2x^3/2, starting from the point P(16, 128)?

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Finding the Arc Length Function for the Curve

To determine the arc length of a curve defined by a function, we can use the arc length formula. For a function y = f(x), the formula for the arc length L from point x = a to x = b is given by:

L = ∫ab √(1 + (dy/dx)2) dx

For our specific case, the function is:

y = 2x3/2

Step 1: Calculate the derivative dy/dx

First, we need to find the derivative of the function:

dy/dx = d/dx(2x3/2) = 2 * (3/2) * x(3/2 – 1) = 3x1/2

Step 2: Substitute into the arc length formula

Now, we need to substitute this derivative into the arc length integral:

1 + (dy/dx)2 = 1 + (3x1/2)2 = 1 + 9x

Now we rewrite the arc length formula:

L = ∫ab √(1 + 9x) dx

Step 3: Determine limits of integration

In this problem, we start from the point P(16, 128). Therefore, the limits of integration will be from:

  • a = 0 (arbitrary starting point)
  • b = 16 (the x-coordinate of point P)

Step 4: Evaluate the integral

The arc length L from 0 to 16 is:

L = ∫016 √(1 + 9x) dx

To solve this integral, you can use substitution:

  1. Let u = 1 + 9x, then du = 9dx or dx = (1/9)du.
  2. Change limits accordingly: when x = 0, u = 1; when x = 16, u = 145.

The integral changes to:

L = (1/9) ∫1145 √u du

Solving this will yield the arc length function between these points. The arc length expression can further be simplified or approximated to provide a numerical value, but this is the general route taken to find the arc length function.

Result

Once evaluated, you will have the arc length related to the interval specified, which provides insight into the distance along the curve from the starting point up to the defined limit.

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