Additional remarks about the areas of plane figures
a) The case of nonpositive function. The area of a figure
represented of the fig. 2, equals
. ( 4 )
b) The case when a function has different signs on various intervals. Let a function is nonnegative on the interval and nonpositive on the interval . The area of a corresponding figure, represented on the fig. 3, equals
. ( 5 )
c) The case when a figure lies between two curves. The area of a figure
represented on the fig. 4, equals
. ( 6 )
Ex. 5. Find the area enclosed by two curves (fig. 5).
Intersection points of the curves have abscises . The figure is symmetric about the Oy-axis; we may find double area of its right part.
Fig. 5 .
d) Figures oriented with respect to the Oy-axis. Areas of figures
(see fig. 6),
(see fig. 7),
(see fig. 8),
(see fig. 9),
are equal respectively
, ( 7 )
, ( 8 )
, ( 9 )
. ( 10 )
Ex. 6. Find the area of a figure bounded by curves (fig. 10).
It’s well to write the equations of the curves in the next form
and to utilize the formula (10).
e) The case of parametrically represented curve.
Let, for example, be given a curvilinear trapezium
but a curve is determined by parametric equations
Ex. 7. Find the area of the loop of a curve (fig. 11).
To construct the curve point by point and to see the loop we equate to zero the expressions , and then form the next table
From the fig. 11 we see that , and so
f) The area in polar coordinates.
Let be given a curvilinear sector (or a curvilinear triangle) that is a plane figure, bounded by two rays and a curve given in polar coordinates (fig. 12). Find the area of the sector.
An element of the area is the area of a hatched circular sector with the radius and a central angle ,
Adding all these elements from to we get the area in question
. ( 11 )
Ex. 8. Find the area of a figure bounded by a cardioid (fig. 13).
Desired area equals the twofold area of upper part of the figure, which is a circular sector with the rays .
Ex. 9. Find the area of a figure bounded by Bernoulli lemniscate
The figure is symmetric with respect to the Ox-, Oy-axes and lies in the angles determined by the straight lines (fig. 14). Passing to polar coordinates , we write the equation of the lemniscate in more suitable form
Then we calculate the quadruplicated area of the circular sector bounded by the lemniscate and the rays . We obtain