Determine the area under the included between the lines x = 0 and x = 1

Given equations are :

...... (1)

x = 0 (y – axis)

x = 1 (represents a line parallel to y - axis at a distance 1 to the right)

equation (1) represents a half eclipse that is symmetrical about the x - axis and also about the y - axis with center at origin.

A rough sketch is given as below: -

We have to find the area of shaded region.

Required area

= (shaded region OABCO)

(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 under the included between the lines x = 0 and x = 1 is equal to square units.

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