Area Between Two Curves

Type-1:

Area between two curves = ∫a[f(x) – g(x)] dx

Type-2:

 

Example:  

Find the area of the region bounded by the parabolas y = x2 and x = y2.

Solution:

When the graph of both the parabolas is sketched we see that the points of intersection of the curves are (0, 0) and (1, 1) as shown in the figure below.

So, we need to find the area enclosed between these points which would give us the area between two curves. Also, in the given region as we can see,

y = x= g(x)

and

x = y2

or, y = √x = f(x).

As we can see in the given region,

The area enclosed will be given as,

Example :

Find the area bounded by curves (x – 1)2 + y2 = 1 and x2 + y2 = 1.

Solution:

Given equations of curves:

x2 + y2 = 1 ….(i)

(x – 1)2 + y2 = 1 ….(ii)

From (i),

y2 = 1 – x2

By substituting it in equation (2), we get;

(x – 1)2 + 1 – x2 = 1

On further simplification

(x – 1)2 – x2 = 0

Using the identity a2 – b2 = (a – b)(a + b),

(x – 1 – x) (x – 1 + x) = 0

-1(2x – 1) = 0

– 2x + 1 = 0

2x = 1

x = 1/2

Using this in equation (1) we get;

y = ± √3/2

Thus, both the equations intersect at point A (1/2, √3/2) and B (1/2, -√3/2).

Also, (0, 0) is the centre of first circle and radius 1

Similarly, (1, 0) is the centre of second circle and radius is 1.

Here, both the circles are symmetrical about x-axis and the required area is shaded here.

So, the required area = area OACB

= 2 (area OAC)

= 2 [area of OAD + area DCA]