Q.233. Area under a parabola

Question 233.
P(at^2,2at) is a point on the parabola y^2=4ax and S is the focus.
Show that the area bounded by the parabola,its axis and the line PS is
(1/3)a^2(3t+t^3).

Answer 233.
X-axis is the axis of the parabola.
Let N be the foot of perpendicular from P on the x-axis
=> N = (at^2, 0)

Area between the parabola, x-axis and the line PS
= Area under the parabola between the x-axis and the line PN - area of the triangle PNS
= ∫ (0 to at^2) ydx - (1/2) * PN * NS
= 2 a^(1/2) ∫ (0 to at^2) x^(1/2) dx - (1/2) * (2at) * (at^2 - a)
= 2 a^(1/2) * [(2/3) x^(3/2)] (x=0 to at^2) - a^2 t^3 + a^2 t
= (4/3) a^2 t^3 - a^2 t^3 + a^2 t
= (1/3) a^2 t^3 + a^2 t
= (1/3) a^2 (3t + t^3).

Link to YA!