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Question
Find the area of the region bounded by the curves x2 = 8y, y = 2, y = 4 and the Y-axis, lying in the first quadrant
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Solution
Given equation of the parabola is x2 = 8y
∴ x = `+- 2 sqrt(2y)`
∴ x = `2sqrt(2y)` .....[∵ In first quadrant, x > 0]
∴ Required area = `int_2^4 x "d"y`
= `int_2^4 2sqrt(2y) "d"y`
= `2sqrt(2)[(y^(3/2))/(3/2)]_2^4`
= `(4sqrt(2))/3 [(4)^(3/2) - (2)^(3/2)]`
= `(4sqrt(2))/3 (8 - 2sqrt(2))`
= `(8sqrt(2))/3 (4 - sqrt(2))` sq.units
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