Q. 135.0( 1 Vote )

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

Answer :

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: -


11.PNG


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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