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