Area

Asked by  | 4th Jan, 2010, 06:59: PM

Expert Answer:

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 - 2x3/2/(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 - 2x3/2/(3/2)

= (-5)(1/2)(1 - cos2y)dy - 2x3/2/(3/2)

= (-5/2) (y - (sin2y)/2) - 2x3/2/(3/2)

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

Regards,

Team,

TopperLearning.

Answered by  | 4th Jan, 2010, 07:56: PM

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