Parametric functions show a relationship between a variable and time
The formula for a parametric equation is given by:
X & Y are dependent and time is independent
Arc length is the distance along a curve
We have to square the derivative, but because both X & Y are dependent, we square both derivatives
We then take the square root and the integral
These functions map numbers to vectors!
For parametric equations, they represent position, velocity, and acceleration!
To derive these functions we have to take the derivative of each component individually
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The same applies to integration- integrate each component individually
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This is a coordinate system where x,y pairs are replaced with r (the distance from the orgin) and θ (angle from x-axis)
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To go from a set of polar coordinates to regular (cartesian) there is a conversion:
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Remember that when we differentiate with polar coordinates we differentiate with respect to theta → dθ
Area between two curves in polar coordinates is given by A = (1/2) ∫(a,b) (R^2 - r^2) dθ
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