How to use this calculator
Choose parametric or polar, enter the curve, the parameter value of the point and the interval for arc length and area. Pi is allowed in every box.
Formulas
Slopedy/dx = (dy/dt)/(dx/dt)
Concavityd²y/dx² = [d/dt(dy/dx)]/(dx/dt)
Arc lengthL = ∫ √((x′)² + (y′)²) dt = ∫ √(r² + (r′)²) dθ
Polar areaA = ½∫ r² dθ
Polar slopedy/dx = (r′ sin θ + r cos θ)/(r′ cos θ − r sin θ)
Worked example
Cycloid x = t − sin t, y = 1 − cos t at t = π/3
- dx/dt = 1 − cos t = 0.5, dy/dt = sin t = 0.866.
- dy/dx = 0.866/0.5 = √3 ≈ 1.732.
- d²y/dx² = −1/(1 − cos t)² = −4: concave down.
- One arch (0 to 2π) has arc length 8 and area 3π.
Frequently asked questions
Why isn’t d²y/dx² just y″/x″?
Because the second derivative is with respect to x, not t. Differentiate dy/dx with respect to t, then divide by dx/dt once more.
Where are the horizontal and vertical tangents?
Horizontal where dy/dt = 0 (and dx/dt ≠ 0); vertical where dx/dt = 0 (and dy/dt ≠ 0). If both vanish, look at the limit of the slope.
Why is my polar area too big?
If the interval traces the curve twice (a rose r = sin 3θ repeats after π), the area is counted twice. Use one trace.