Given equations are:
x = 2 ...... (1)
And y2 + 1 = x, x ≤ 2 ...... (2)
equation (2) represents a parabola with vertex at (1, 0) and passing through (2, 0) on x - axis, equation (1) represents a line parallel to y - axis at a distance of 2 units.
A rough sketch is given as below: -
We have to find the area of shaded region.
Required area
= shaded region ABCA
= 2 (shaded region ACDA) ( as it is symmetrical about the x - axis)
(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)
\(=2\int^2_1 y\,dx\) (As x is between (1, 2) and the value of y varies)
Substitute u = x - 1 \(\Rightarrow\) dx = du
So the above equation becomes,
On integrating we get,
On applying the limits we get,
Hence the area enclosed by the curve and the line x = 2 is equal to \(\frac{4}{3}\) square units.
