Draw the rough sketch of y^{2} + 1 = x, x ≤ 2. Find the area enclosed by the curve and the line x = 2

Given equations are:

x = 2 ...... (1)

And y^{2} + 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)

(As x is between (1,2) and the value of y varies)

(as )

Substitute

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 square units.

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