Generated 2/24/2026, 1:43:37 AM
The equation for arc length given y = f(x) is “∫ √(1 + f’(x)²) dx”. Would the quation for arc lenght given x = f(y) be “∫ √(1 + f’(y)²) dy”?
Yes, you are correct. When the curve is given as x = f(y) instead of y = f(x), the arc length formula becomes:
L = \int_c^d \sqrt{1 + \left[f'(y)\right]^2} \, dy
where f'(y) = \frac{dx}{dy} and the integration is performed from y = c to y = d.nd+1
The arc length formula for y = f(x) is derived from the Pythagorean theorem applied to small segments of the curve, giving \sqrt{(\Delta x)^2 + (\Delta y)^2}, which can be rewritten as \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx. When the curve is instead described as x = f(y), the roles of x and y are reversed, and we factor out dy instead of dx, leading to \sqrt{1 + \left(\frac{dx}{dy}\right)^2} \, dy.courses.lumenlearning+1
The key difference is that for x = f(y), you’re integrating with respect to y and using the derivative \frac{dx}{dy}, whereas for y = f(x), you integrate with respect to x and use \frac{dy}{dx}.