##### 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

(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 ) Substitute 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.

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