Question

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

Answer 1

Given parabolas are y² = 2px …. (i) and x² = 2py …. (ii)

Now, from equation (ii) we have

y = x²/2p

Putting the value of y in equation (i), we have

(x²/2p)² = 2px

x⁴/4p² = 2px

x⁴ = 8p³x

x⁴ – 8p³x = 0

x(x³ – 8p³) = 0

So, x = 0 or x³ – 8p³ = 0 ⇒ x = 2p

Now, the required area is

Therefore, the required area is 4/3 p² sq. units.