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Draw a rough sketch of the graph of the curve and evaluate the area of the region under the curve and above the x - axis
Given equations are:
...... (1)
And x - axis ...... (2)
equation (1) represents an eclipse that is symmetrical about the x - axis and also about the y - axis, with center at origin and passes through (±2, 0) and (0, ±3).
A rough sketch is given as below: -
We have to find the area of shaded region.
Required area
= shaded region ABCA
= 2 (shaded region OBCO) ( 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 (0,2) 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 of the region under the given curve and above the x - axis is equal to square units.