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1. The area region bounded by the parabolas y2=4ax and x2=4ay is
We have, y2 = 4ax --------------------------- (1) x2 = 4ay ---------------------------- (2) (1) and (2) intersects hence x = y2/4a (a > 0) => (y2/4a)2 = 4ay => y4 = 64a3y => y4 – 64a3y = 0 => y[y3 – (4a)3] = 0 => y = 0, 4a When y = 0, x = 0 and when y = 4a, x = 4a. The points of intersection of (1) and (2) are O(0, 0) and A(4a, 4a). The area of the region between the two curves = Area of the shaded region = 0∫4a(y1 – y2)dx = 0∫4a[√(4ax) – x2/4a]dx = [2√a.(x3/2)/(3/2) – (1/4a)(x3/3)]04a = 4/3√a(4a)3/2 – (1/12a)(4a)3 – 0 = 32/3a2 – 16/3a2 = 16/3a2 sq. units
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