Q. 85.0( 1 Vote )

Evaluate the area bounded by the ellipse above the x-axis.

Answer :

Given the boundaries of the area to be found are,


• The ellipse,


• y = 0 (x-axis)


From the equation, of the ellipse


• the vertex at (0,0) i.e. the origin,


• the minor axis is the x-axis and the ellipse intersects the x- axis at A(-2,0) and B(2,0).


• the major axis is the y-axis and the ellipse intersects the y- axis at C(3,0) and D(-3,0).



As x and y have even powers, the area of the ellipse will be symmetrical about the x-axis and y-axis.


Here the ellipse, , can be re-written as




----- (1)


As given, the boundaries of the re to be found will be


• The ellipse, with vertex at (0,0).


• The x-axis.


Now, the area to be found will be the area under the ellipse which is above the x-axis.


Area of the required region = Area of ABC.


Area of ABC = Area of AOC + Area of BOC


[area of AOC = area of BOC as the ellipse is symmetrical about the y-axis]


Area of ABC = 2 Area of BOC




[Using the formula, ]




[sin-1(1) = 90° and sin-1(0) = 0° ]


The Area of the required region 3π sq. units


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