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

Answer 1

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