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Find the area under the curve above x - axis from x = 0 to x = 2. Draw a sketch of curve also.
Given equations are:
x – axis ...... (1)
x = 0 ...... (2)
x = 2 ...... (3)
And ...... (4)
equation (4) represents a half parabola with vertex at and passing through (2,0) on x - axis, equation (3) represents a line parallel to y - axis at a distance of 2 units and equation (2) represents y - axis.
A rough sketch is given as below: -
We have to find the area of shaded region.
= shaded region OABC
(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)
So the above equation becomes,
On integrating we get,
On applying the limits we get,
Hence the area under the curve above x - axis from x = 0 to x = 2 is equal to square units.