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Interactive Calculus Visualizations

Interactive Calculus Visualizations













Calculus Concepts Visualized

An interactive exploration of parametric and polar formulas.






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)