Q. 135.0( 1 Vote )

Determine t

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

(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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