Question #346267

Using shell method, find the volume of R, when it is bounded by √x + √y = √a , x =

0 , y = 0 about the line x = a.


1
Expert's answer
2022-05-30T23:01:40-0400


x+y=a,\sqrt{x}+\sqrt{y}=\sqrt{a},

x=(ay)2=a2ay+yx=(\sqrt{a}-\sqrt{y})^2=a-2\sqrt{a}\sqrt{y}+y




V=0a2πy(a(a2ay+y))dyV=\displaystyle\int_{0}^a2\pi y(a-(a-2\sqrt{a}\sqrt{y}+y))dy

=2π0a(2ay3/2y2)dy=2\pi\displaystyle\int_{0}^a(2\sqrt{a}y^{3/2}-y^2)dy

=2π[4ay5/25y33]a0=2\pi[\dfrac{4\sqrt{a}y^{5/2}}{5}-\dfrac{y^3}{3}]\begin{matrix} a \\ 0 \end{matrix}

=14πa315(units3)=\dfrac{14\pi a^3}{15} ({units}^3)

14πa315\dfrac{14\pi a^3}{15} cubic units.


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