Interactive Calculus Visualizations

Parametric Derivative (Slope)

Finds the slope of the tangent line for a parametric curve. Remember this as the change in y over the change in x, both with respect to the parameter t.

$$ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} $$

Parameter 't': 1

Curve: $ x(t) = t^2 - 2, y(t) = t^3 - 3t $

Slope at t=1 is -

Parametric Arc Length

Calculates the length of a curve defined parametrically from t=a to t=b.

$$ L = \int_{a}^{b} \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} \,dt $$

Start 'a': -1.5

End 'b': 1.5

Arc Length: -

Polar Area

Finds the area of a region bounded by a polar curve. Remember the $ \frac{1}{2} $ and that the radius is squared.

$$ A = \int_{\alpha}^{\beta} \frac{1}{2} r^2 \,d\theta $$

Start Angle α: 0°

End Angle β: 90°

Curve: $ r(\theta) = 3 \cos(4\theta) $

Area: -

Polar Arc Length

Calculates the length of a curve defined in polar coordinates from θ=a to θ=b.

$$ L = \int_{a}^{b} \sqrt{r^2 + (\frac{dr}{d\theta})^2} \,d\theta $$

Start Angle a: 0°

End Angle b: 180°

Polar Arc Length: -

Coordinate Conversion

Convert coordinates between Polar and Cartesian systems.

Polar to Cartesian

$$ x = r \cos(\theta), \quad y = r \sin(\theta) $$

Radius (r): 2

Angle (θ): 45°

Cartesian: (x, y)

Cartesian to Polar

$$ r^2 = x^2 + y^2, \quad \tan(\theta) = \frac{y}{x} $$

X-coordinate: 1

Y-coordinate: 1

Polar: (r, θ)