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Question

The value of integral over the region in positive quadrant for which x+y1 is ______
  1. 0.16

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Solution

The correct option is A 0.16
The region of integration R is the area bounded by the straight lines x+y=1, and the co-ordinate axes i.e it is bounded by the lines.
x=0,x=1y,y=0, and y=1

Integrating first over the horizontal strip PQ i.e w.r.t from the point P(x=0) to Q(x=1y) and then w.r.t y from y=0 to y=1 we get

R(x2+y2)dxdy=101y0(x2+y2)dxdy

=10[x33+xy2]1y0dy=10[(1y)33+(1y)y2]dy

=[(1y)412y33y44]10=(1314)(112)=16=0.1667

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