The areas of plane figures
If a continuous curve is defined in rectangular coordinates by the equation
points x=a and x =b and by a segment of the x-axis
In the more general case, if the area S is bounded by two continuous curves
If the curve is defined by equations in parametric form
where If a curve is defined in polar coordinates by the equation
Fig. 2. which correspond to the values
|
the area of the curvilinear trapezoid bounded by this curve, by two vertical lines at the
, is given by the formula
.
and 
and by two vertical lines x=a and x=b, where 
when 
.
and 
then the area of the curvilinear trapezoid bounded by this curve, by two vertical lines (x=a and x=b), and by a segment of the x-axis is expressed by the integral
,
and 
are determined from the equations 
and 
on the interval 
.
, then the area of the sector AOB (Fig. 2), bounded by an arc of the curve, and by two radius vectors OA and OB,
and 
is expressed by the integral
.
