Find the area bounded by the parabola `y^2 = 4ax` and the line `y = 2ax`.

Explanation

`y^(2)=4aximpliesy=2sqrt(a)sqrt(x) " " `...(1)
`y=2ax " " ` ...(2)
On solving the eqs. (1) and (2), we get the points of intersection are O(0 , 0) and `P((1)/(a), 2)`
Draw PQ `bot` OX

Now the area of the region bounded by the parabola and the straight line
`=` Area of OQPAO - Area of OQPBO
`=int_(0)^(1//a)sqrt(a)sqrt(x)dx-int_(0)^(1//a)2axdx`
`=2sqrt(a).[(2)/(3)x^(3//2)]_(0)^(1//a)-2a[(x^(2))/(2)]_(0)^(1//a)`
`=(4)/(3)sqrt(a).(1)/(a^(3//2))-a.(1)/(a^(2))`
`=(4)/(3a)-(1)/(a)=(1)/(3 a)sq.` units.