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Question
Find the area of the region bounded by the parabola y2 = 2px, x2 = 2py
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Solution
We are given that: x2 = 2py ......(i)
And y2 = 2px ......(ii)
From equation (i)
We get y = `x^2/(2"p")`
Putting the value of y in equation (ii)
We have `(x^2/(2"p"))` = 2px
⇒ `x^4/(4"p"^2)` = 2px
⇒ x4 = 8p3x
⇒ x4 – 8p3x = 0
⇒ x(x3 – 8p3) = 0
∴ x = 0, 2p
Required area = Area of the region (OCBA – ODBA)
= `int_0^(2"p") sqrt(2"p"x) "d"x - int_0^(2"p") x^2/(2"p") "d"x`
= `sqrt(2"p") * 2/3 [x^(3/2)]_0^(2"p") - 1/(2"p") * 1/3 [x^3]_0^(2"p")`
= `(2sqrt(2))/3 sqrt("p") [(2"p")^(3/2) - 0] - 1/(6"p") [(2"p")^3 - 0]`
= `(2sqrt(2))/3 sqrt("p") * 2sqrt(2) "p"^(3/2) - 1/(6"p") * 8"p"^3`
= `8/3 * "p"^2 - 8/6 "p"^2`
= `8/6 "p"^2`
= `4/3 "p"^2` sq.units
Hence, the required area = `4/3 "p"^2` sq.units
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