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**Solve the following :**

Find the area of the region bounded by the curve y = 4x^{2}, Y-axis and the lines y = 1, y = 4.

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#### Solution

By symmetry of the parabola, the required area is 2 times the area of the region ABCD.

From the equation of the parabola, x^{2} = `y/(4)`

the first quadrant, x > 0

∴ x = `(1)/(2)sqrt(y)`

∴ required area = `int_1^4 x*dy`

= `(1)/(2) int_1^4 sqrt(y)*dy`

= `(1)/(2)[y^(3/2)/(3/2)]_1^4`

= `(1)/(2) xx (2)/(3)[4^(3/2) - 1^(3/2)]`

= `(1)/(3)[(2^2)^(3/2) - 1]`

= `(1)/(3)[8 - 1]`

= `(7)/(3)"sq units"`.

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