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Question
Solve the following:
Find the area of the region bounded by the curve y = 4x2, 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, x2 = `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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