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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 cycloid x = 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

 


  this page updated 21-feb-16
Mathwords: Terms and Formulas from Algebra I to Calculus
written, illustrated, and webmastered by Bruce Simmons
NCTM Web Bytes December 2004 Web Bytes March 2005 Web Bytes