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 :
$S = - α \int β y (t) x^{'} (t) d t,$
$S = α \int β x (t) y^{'} (t) d t,$
$S = \frac{1}{2} α \int β (x y^{'} - y x^{'}) d t,$
Where $α and β$ 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.
The area of the loop described as
$x = \frac{t}{3} (6 - t), y = \frac{t ^{2}}{8} (6 - t)$ is

Question

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 \int β y (t) x^{'} (t) d t,

S = α \int β x (t) y^{'} (t) d t,

S = \frac{1}{2} α \int β (x y^{'} - y x^{'}) d t,

Where

α and β

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.
The area of the loop described as

x = \frac{t}{3} (6 - t), y = \frac{t ^{2}}{8} (6 - t)

is

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