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Question

Calculate the area of the region bounded by the parabolas y2 = x and x2 = y.

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Solution



y2=x is a parabola, opening sideways, with vertex at O(0, 0) and +ve x-axis as axis of symmetry x2 =y is a parabola, opening upwards, with vertex at O(0, 0) and +ve y-axis as axis of symmetry Soving the above two equations,x2=y4=y y4-y =0 y=0 or y=1 . So, x=0 or x=1O0, 0 and A(1, 1) are points of intersection of two curvesConsider a vertical strip of length = y2-y1 and width= dx Area of approximating rectangle =y2-y1 dx Approximating rectangle moves from x=0 to x=1Area of the shaded region =01y2-y1 dx A=01 y2-y1 dx As, y2-y1>0 y2-y1=y1A=01 x -x2 dx A=x3232-x3301A=23×132-133 -0A=23-13A=13 sq. units Thus, area enclosed by the curves =13 sq. units

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