Given equation:
equation (1) represents a half eclipse that is symmetrical about the x - axis and also about the y - axis with center at origin and passes through (±1, 0) and (0, ±2). And x ∈ [0, 1] is represented by region between y - axis and line x = 1.
A rough sketch is given as below: -
We have to find the area of shaded region.
Required area
= (shaded region OBCO)
(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)
\(=\int^1_0 y\,dx\) (As x is between (0,1) and the value of y varies)
Substitute x = sin u \(\Rightarrow\) u = sin-1(x), dx = cos u du
So the above equation becomes,
So the above equation becomes,
Apply reduction formula:
Hence the area enclosed between the curve and the x - axis is equal to \(\frac{\pi}{2}\) square units.
