Find the area of the region bounded by the curve x = at2, y = 2at between the ordinates corresponding t = 1 and t = 2
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.
= (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)
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.