Project ForgeEngineering Mathematics
Calculus
GATE 2009
Hard
A path AB in the form of one quarter of a circle of unit radius is shown in the figure, from B=(0,1) to A=(1,0). Integration of (x+y)² on path AB traversed in a counter-clockwise sense is
- Aπ/2-1
- π/2+1
- Cπ/2
- D1
Explanation
Parametrize x=cosθ, y=sinθ, ds=dθ, θ from 0 to π/2. (x+y)²=1+2sinθcosθ=1+sin2θ. ∫0^(π/2)(1+sin2θ)dθ=[θ-cos2θ/2] from 0 to π/2=(π/2+1/2)-(-1/2)=π/2+1.
Source: GATE ME 2009 (2009)
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