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in Definite Integrals by (28.7k points)
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Prove that the area common to the two parabolas y = 2x2 and y = x2 + 4 is 32/3 sq. Units.

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To find the area enclosed by,

y = 2x2 ...(i)

And y = x2 + 4 ...(ii)

On solving the equation (i) and (ii),

2x2 = x2 + 4

Or x2 = 4

Or x = \(\pm 2\)

\(\therefore\) y = 8

Equation (1) represents a parabola with vertex (0, 0) and axis as y - axis.

Equation (2) represents a parabola with vertex (0,4) and axis as the y - axis.

Points of intersection of parabolas are A(2, 8) and B(– 2, 8).

These are shown in the graph below: -

Required area = Region AOBCA

= 2(Region AOCA)

Hence, proved that the area common to the two parabolas y = 2x2 and y = x2 + 4 is 32/3 sq. Units.

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