Question

How do you do this question?

Answer 1

ds = √(dx² + dy²)

ds = √(1 + (dy/dx)²) dx

ds = √(1 + (dx/dy)²) dy

Explanation:

Arc length ds is found with Pythagorean theorem:

ds² = dx² + dy²

Solving for ds:

ds = √(dx² + dy²)

If we factor out dx from the radical:

ds = √(dx² (1 + (dy/dx)²))

ds = √(1 + (dy/dx)²) dx

Or, if we factor out dy from the radical:

ds = √(dy² ((dx/dy)² + 1))

ds = √(1 + (dx/dy)²) dy