Given equations are:

x = at² ...... (1)

y = 2at ..... (2)

t = 1 ..... (3)

t = 2 ..... (4)

Equation (1) and (2) represents the parametric equation of the parabola.

Eliminating the parameter t, we get

This represents the Cartesian equation of the parabola opening towards the positive x - axis with focus at (a,0).

A rough sketch of the circle is given below: -

When t = 1, x = a

When t = 2, x = 4a

We have to find the area of shaded region.

Required area

= (shaded region ABCDEF)

= 2(shaded region BCDEB)

(the area can be found by taking a small slice in each region of width Δx, then the area of that sliced part will be yΔx as it is a rectangle and then integrating it to get the area of the whole region)

(As x is between and the value of y varies, here y is Cartesian equation of the parabola)

(as )

On integrating we get,

(by applying power rule)

On applying the limits we get,

Hence the area of the region bounded by the curve x = at², y = 2at between the ordinates corresponding t = 1 and t = 2 is equal to square units.