CBSE Class 12 Mathematics Application Of Integration Notes

(A) KEY CONCEPTS

1. AREA LYING BELOW THE X-AXIS:

If f(x)≤0 for a≤x≤b,then the graph of y=f(x) lies below x-axis Therefore area bounded by the curve y=f(x),x-axis and the ordinates x=a and x=b is given by

2. AREA LYING ABOVE THE X-AXIS:

The area enclosed by the curve y= f(x), x-axis & between the ordinate at x=a & x=b is given

3. AREA LYING ON RIGHT OF Y-AXIS :

Area bounded by the curve x=f(y),y-axis and the abscissa y=c and y=d is given by

4. AREA LYING ON LEFT OF Y-AXIS:

The area enclosed by the curve x= f(y), y-axis & between the abscissa at y=c & y=d is given by :

5. AREA BOUNDED BY TWO CURVES

Area bounded by the two curves y = f(x) & y = g(x) where f1(x) f2(x) in  a , b  & between the ordinate x=a & x=b is given by

 IMPORTANT FORMULAE TO USE :

Important Notes

1. If the equation of the curve contains only even powers of x, then the curve is symmetrical about y-axis

2. If the equation of the curve contains only even powers of y, then the curve is symmetrical about x-axis.

3. If the equation of the curve remains unchanged when x is replaced by –x and y by –y, then the curve is symmetrical in opposite quadrants.

4. If the equation of the curve remains unchanged when x and y are interchanged ,then the curve is symmetrical about the line y=x

 1. Find the area of the region {(x,y):x² ≤ y ≤ x }

Sol. The required area is bounded between two curves y =x² and y= x . Both of these curves are symmetric about y-axis and shaded region in the fig. shows the region whose area is required.

Therefore, required area =2× area of region R1

Now to find point of intersection of curves y =x² and y= x , we solve them simultaneously.

Clearly, region R1 is in first quadrant, where x>0

x =x => y =x…………….(i)

y =x²…………….(ii)

either x = 0 or x = 1

The limits are , when x=0, y=0 and when x=1, y=1

So points of intersection of the curve are o(0,0) and A(1,1)

Now, required area = 2× area of region R₁

Please click the link below to download CBSE Class 12 Mathematics Application of Integration.