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in Definite Integrals by (34.5k points)
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Find the area of the region bounded by the parabola y2 = x and the line y = x in the first quadrant.

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To obtain the points of intersection of the line and the parabola, we equate the values of x from both equations

∴ y2 = y

∴ y2 – y = 0

∴ y(y – 1) = 0

∴ y = 0 or y = 1

When y = 0, x = 0

When y = 1, x = 1

∴ the points of intersection are O(0, 0) and A(1, 1).

Required area = area of the region OCABO = area of the region OCADO – area of the region OBADO

Now, area of the region OCADO = area under the parabola y2 = x i.e. y = +√x (in the first quadrant) between x = 0 and x = 1

Area of the region OBADO = area under the line y = x between x = 0 and x = 1

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