Area Types of Problems
We know the method(s) of finding areas of triangle,
quadrilateral, polygon, circle etc. In this chapter we shall learn how to find
the area of the regions bounded by curves of the following types.
Type I: y = f(x), the x-axis and the lines x = a and x
= b where f is a continuous function in [a, b].
Type II: The curves y = f1(x), x ∈
(a, c), y = f2(x) , x ∈ (c, b), the x-axis and the lines x= a, x
= b ; a < c< b . .
Type III: The curves y = f1(x), y = f2(x) and the
lines x = a, x = b.
Type IV: Bounding curves are represented by function
defined through given conditions.
Type V: Miscellaneous
Type I:
(i) The area bounded by y = f(x) above the x- axis
and the coordinates x = a and x = b (represented by shaded portion ) is given
by
![](http://www.quizsolver.com/radix/dth/notif/Area%20Types%20%20of%20Problems_files/image001.gif)
![](http://www.quizsolver.com/radix/dth/notif/Area%20Types%20%20of%20Problems_files/image002.gif)
Example.1
Calculate the area bounded by the parabola y = x2,
x-axis, and the lines x =2, x=4. .
Solution:
=
square
units.
Type II:
The area of the shaded portion is given by
, where x = c is a
solution of f1(x) = f2(x), a< c< b.
![](http://www.quizsolver.com/radix/dth/notif/Area%20Types%20%20of%20Problems_files/image006.gif)
Example.1
Find the area included between the curves y = sin-1 x ,
y = cos-1x and the x-axis.
Solution:
Clearly we have to find the area of the shaded region
OPBO. The point P of intersection of .
y = sin-1x and y = cos-1x is obtained by solving
sin-1x = cos-1x
&⇒ p/2 - cos-1x = cos-1x
&⇒ 2 cos-1x = p/2
&⇒ cos –1x = p/4
x =
![](http://www.quizsolver.com/radix/dth/notif/Area%20Types%20%20of%20Problems_files/image008.gif)
The required area
D = ![](http://www.quizsolver.com/radix/dth/notif/Area%20Types%20%20of%20Problems_files/image009.gif)
=
=
![](http://www.quizsolver.com/radix/dth/notif/Area%20Types%20%20of%20Problems_files/image012.gif)
Put x = sinq
dx = cosq dq
x = 0, q = 0
x = 1/
, q = p/4
x = 1, q = p/2
&⇒ The required area
D =
= ![](http://www.quizsolver.com/radix/dth/notif/Area%20Types%20%20of%20Problems_files/image015.gif)
= ![](http://www.quizsolver.com/radix/dth/notif/Area%20Types%20%20of%20Problems_files/image016.gif)
=
sq. units.
Type III:
The shaded area, which is bounded by two curves y = f1
(x), y = f2(x) and the ordinates
x = a, x= b is given by
.(*)
Note: If y= f1(x) is above the x-axis and
y = f2(x) is below the x-axis, or y = f1(x) and
y = f2(x) are both below the x-axis or partly above
and partly below the x-axis, even then formula (*) works. .
![](http://www.quizsolver.com/radix/dth/notif/Area%20Types%20%20of%20Problems_files/image019.gif)
Example.1
Find the area included between the line y = x and the
parabola x2 = 4y.
Solution:
The line y = x cuts the parabola at O and B whose x
coordinates are 0 and 4 respectively
The required area = ![](http://www.quizsolver.com/radix/dth/notif/Area%20Types%20%20of%20Problems_files/image020.gif)
=
=![](http://www.quizsolver.com/radix/dth/notif/Area%20Types%20%20of%20Problems_files/image022.gif)
=
=
sq. units.
![](http://www.quizsolver.com/radix/dth/notif/Area%20Types%20%20of%20Problems_files/image025.gif)
Type IV:
In this type we consider those cases, when the function
is not given explicitly. Functions representing the curves are to be obtained
or formulated from the given condition. This is illustrated by the following
example. .
Example.1
Find the area bounded by f(x) = maximum{sinx, cosx},
x= 0 , x = 2p and the x-axis.
Solution:
The graph of the function y = f(x) is shown by bold
curves. .
The required region is shaded.
The required area
=![](http://www.quizsolver.com/radix/dth/notif/Area%20Types%20%20of%20Problems_files/image026.gif)
![](http://www.quizsolver.com/radix/dth/notif/Area%20Types%20%20of%20Problems_files/image027.gif)
![](http://www.quizsolver.com/radix/dth/notif/Area%20Types%20%20of%20Problems_files/image029.gif)
=
= Ö2+
4 sq. units.
Type-V:
In this type we consider those cases when a variable
area is bounded by different curves (One curve unknown) is given, and we have
to find the unknown curve y = f(x)
Example.1
If the area bounded by the x-axis, the curve y = f(x)
and the lines x = a, x = b is equal to
, " a
< b, where ‘a’ is a given positive real number, find f(x).
Solution:
According to the given condition ![](http://www.quizsolver.com/radix/dth/notif/Area%20Types%20%20of%20Problems_files/image032.gif)
Differentiating with respect to b (as b is a variable),
we get
f(b) = ±
.2b
f(x) = ±![](http://www.quizsolver.com/radix/dth/notif/Area%20Types%20%20of%20Problems_files/image034.gif)