CBSE Class 12-science Answered

x²+y² = 5 and y² = 4x

The intersection coordinates,

x²+4x = 5

x = 1,

(x, y) = (1,2)

Area of smaller region bounded by the curves =

Area under the circular arc from (-1, 2) to (1,2) - Area under the parabola from (-1, 2) to (1,2)

= (ydx)_{Circular Arc} - (ydx)_Parabola

= (5-x²)^1/2 dx - (4x)^1/2 dx

= (5)^1/2(1-cos²y)^1/2((5)^1/2(-siny))dy - 2x^3/2/(3/2)

{ Put x = (5)^1/2cosy, then dx = -(5)^1/2siny and the limits are, x = 1, y = cos^-1(1/5), x = 1, y = cos^-1(-1/5) for the first integral and second integral is simple.}

= (5)(-sin²y)dy - 2x^3/2/(3/2)

= (-5)(1/2)(1 - cos2y)dy - 2x^3/2/(3/2)

= (-5/2) (y - (sin2y)/2) - 2x^3/2/(3/2)

Substitute the upper and lower limits, to get the answer.

Regards,

Team,

TopperLearning.