Question

Find the area of the region included between: y² = 4ax and the line y = x

Answer 1

x2 = 4ax

x2 - 4ax = 0

x (x - 4a) = 0

x = 0 or x - 4a = 0

x = 0 or x = 4a

∴ y = x

∴ Points of intersectionre 

(0, 0) and (4a, 4a)

the area bounded by a parabola and line  = (OCBDO)

A = A (OABDO - (ΔOAB)

A = Area undercurve - Area underline

`A = int_0^(4a) 2 sqrt( ax )dx - int_0^(4a) x dx`

`A = 2sqrta [x^(3/2)/(3/2)]_0^(4a) - [x^2/2]_0^(4a)`

`A = 2sqrtaxx 2/3 [(4a)^(3/2) - 0 - 1/2 [(4a)^2 - 0]`

`A = 4^sqrta/3 [8a^(3/2) - 1/2][16a^2]`

`A = 32/3 a^2 - 8a^2`

`A = (32/3-8) a^2`

`A = (32 - 24)/3 a^2`

`A = 8/3 a^2`