Question

Find the area of the region bounded by the parabola y² = 2px, x² = 2py

Answer 1

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