Using double integral fund the area of region enclosed by √x+√y=√a and x+y=a
1
Expert's answer
2022-03-11T05:53:06-0500
Area bounded by the two curves = ∬Rdydx.
The limits of the integral are obtained from the given equations.
x+y=a⟹y=a−x⟹y=(a−x)2x+y=a⟹y=a−x
The limits for x are obtained from the two equations of the curve.
(a−x)2a−2ax+x2x4x24x2−4ax4x(x−a)=a−x=a−x=2ax=4ax(Squaring both sides)=0=0
which gives, x=0&x=a.
Hence, y varies from y=(a−x)2 to y=a−x and x varies from x=0 to x=a.
Therefore,
Area bounded by the curves =∫0a∫(a−x)2a−xdydx=∫0ay∣∣(a−x)2a−xdx=∫0a((a−x)−(a−x)2)dx=∫0a(a−x−a−x+2ax)dx=∫0a(−2x+2ax)dx=2∫0a(ax−x)dx=2(a(23)x23−2x2)∣∣0a=2(32a⋅a23−2a2)=2(32a2−2a2)=3a2
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