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Find the area of the region bounded by the curve y = (x^{2} + 2)^{2}, the X-axis and the lines x = 1 and x = 3

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#### Solution

Let A be the required area.

Given equation of the curve is y = (x^{2} + 2)^{2 }

∴ A = `int_1^3 y "d"x`

= `int_1^3 (x^2 + 2)^2 "d"x`

= `int_0^3 (x^4 + 4x^2 + 4)`

= `[x^5/5 + 4(x^3/3) + 4x]_1^3`

= `[3^5/5 + 4(3^3/3) + 4(3)] - [1^5/5 + 4(1^3/3) + 4(1)]`

= `(243/5 + 36 + 12) - (1/5 + 4/3 + 4)`

= `483/5 - 83/15`

= `(1449 - 83)/15`

= `1366/15` sq.units

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