Q23 of 26 Page 1

Using integration find the area of the region

Given: parabola: and circle


Plotting both the graphs to find the common region



We have to find the area of the part which is common to both circle and parabola. First let us find the intersection points of the curve.


Solving both the equations


Substituting y2 = 6ax in the equation circle,


x2 + 6ax = 16a2


(x - 2a) (x + 8a) = 0


x = 2a or x = - 8a


x = - 8a is not possible as it lies outside the common area.


And putting x = 2a in equation of parabola, we get,


y2 = 6 × a × 2a


y2 = 12a2


y = ±2√2 a


Hence, the intersection points are (2a, +2√2 a) and (2a, -2√2 a)


Now finding the area by integration,




Substituting the limits




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