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area using parametric equations

Area Using Parametric Equations
Parametric Integral Formula

The area between the x-axis and the graph of x = x(t), y = y(t) and the x-axis is given by the definite integral below. This formula gives a positive result for a graph above the x-axis, and a negative result for a graph below the x-axis.

Note: If the graph of x = x(t), y = y(t) is partly above and partly below the x-axis, the formula given below generates the net area. That is, the area above the axis minus the area below the axis.

 

Formula:

       
Example:

Find the area of the between the x-axis and the first period of the cycloidx = t – sin t,y = 1 – cos t. The values of t run from 0 to 2π.

 

See also

Parametrize, parametric equations, area under a curve, area using polar coordinates

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