Q. 39

Make a sketch of the region{(x,y): 0 y x2 + 3; 0 y 2x + 3; 0 x 3} and find its area using integration.

Answer :

To find area given equations are

Y = x2 + 3 …(i)


Y = 2x + 3 …(ii)


And x = 3 …(iii)


Solving the above three equations to get the intersection points,


x2 + 3 = 2x + 3


Or x2 – 2x = 0


Or x(x – 2) = 0


And x = 0 or x = 2


y = 3 or y = 7


Equation (1) represents a parabola with vertex (3, 0) and axis as y – axis.


Equation (2) represents a line a passing through (0, 3) and ( – 3/2, 0)


The points of intersection are A (0,3) and B(2,7).


These are shown in the graph below:



Required area =









The area of the region{(x,y): 0yx2 + 3; 0 y 2x + 3; 0 x 3} is


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