Class 12

Math

Calculus

Area

Computing area with parametrically represented boundaries : If the boundary of a figure is represented by parametric equation, i.e., $x=x(t),y=(t),$ then the area of the figure is evaluated by one of the three formulas :

$S=−α∫ β y(t)x_{′}(t)dt,$

$S=α∫ β x(t)y_{′}(t)dt,$

$S=21 α∫ β (xy_{′}−yx_{′})dt,$

Where $αandβ$ are the values of the parameter t corresponding respectively to the beginning and the end of the traversal of the curve corresponding to increasing t.

If the curve given by parametric equation $x=t−t_{3},y=1−t_{4}$ forms a loop for all values of $t∈[−1,1]$ then the area of the loop is