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Question
Find the area of the region bounded by the curve y2 = 27x3 and the lines x = 0, y = 1 and y = 2
Solution
Equation of the curve is y2 = 27x3
⇒ x³ = `y^2/27 = y^3/3^3`
∴ x = `(y)^(2/3)/3`
Since the Area of the region bounded by the given curve and the lines x = 0, y = 1 and y = 2
∴ Area A = `int_1^2 x "d"y`
= `int_1^2 (y)^(2/3)/3 "d"y`
`1/3 int_1^2 (y)^(2/3) "d"y`
= `1/3[(y)^(2/3 + 1)/((2/3) + 1)]_1^2`
= `1/3 [(y)^(5/3)/((5/3))]_1^2`
= `1/3 xx 3/5 xx [(2)^(5/3) - (1)^(5/3)]`
A = `1/5 ((2)^(5/3) - 1)` sq.units
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