Draw a rough sketch of the graph of the function y=2√1–x2,x[0,1] and evaluate the are enclosed between the curve and the x–axis.

Given equation:

...... (1)

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)

(As x is between (0,1) and the value of y varies)

(as )


So the above equation becomes,

We know,

So the above equation becomes,

Apply reduction formula:

On integrating we get,

Undo the substituting, we get

On applying the limits we get,

Hence the area enclosed between the curve and the x - axis is equal to square units.