Question #308471

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\displaystyle\iint_{R}dydx.


The limits of the integral are obtained from the given equations.

x+y=a    y=ax    y=(ax)2x+y=a    y=ax\sqrt{x} + \sqrt{y} = \sqrt{a} \implies\sqrt{y} = \sqrt{a}-\sqrt{x} \implies y = (\sqrt{a}-\sqrt{x})^2\\ x+y=a \implies y = a-x


The limits for x are obtained from the two equations of the curve.

(ax)2=axa2ax+x=ax2x=2ax4x2=4ax(Squaring both sides)4x24ax=04x(xa)=0\begin{aligned} (\sqrt{a} - \sqrt{x})^{2} &= a - x\\ a - 2\sqrt{ax} + x &= a - x\\ 2x &= 2\sqrt{ax}\\ 4x^{2} &= 4ax \quad\text{(Squaring both sides)}\\ 4x^{2} - 4ax &= 0\\ 4x(x-a) &= 0 \end{aligned}

which gives, x=0 & x=ax = 0~ \&~ x=a.


Hence, y varies from y=(ax)2 to y=axy = (\sqrt{a}-\sqrt{x})^2 \text{ to } y = a-x and x varies from x=0 to x=a.x = 0 \text{ to } x = a.


Therefore,

Area bounded by the curves =0a(ax)2axdydx=0ay(ax)2axdx=0a((ax)(ax)2)dx=0a(axax+2ax)dx=0a(2x+2ax)dx=20a(axx)dx=2(a x32(32)x22)0a=2(23aa32a22)=2(2a23a22)=a23\begin{aligned} \text{Area bounded by the curves } &= \displaystyle\int_{0}^{a}\int_{(\sqrt{a}-\sqrt{x})^2}^{a-x}dydx\\ &= \displaystyle\int_{0}^{a} y\Bigg|_{(\sqrt{a}-\sqrt{x})^2}^{a-x} dx\\ &= \displaystyle\int_{0}^{a} ((a-x)-(\sqrt{a}-\sqrt{x})^2)dx\\ &= \displaystyle\int_{0}^{a} (a - x - a - x +2\sqrt{ax})dx\\ &= \displaystyle\int_{0}^{a} ( - 2x + 2\sqrt{ax})dx\\ &= 2\displaystyle\int_{0}^{a} ( \sqrt{ax} - x)dx\\ &= 2 \Big(\sqrt{a} ~\frac{x^{\frac{3}{2}}}{(\frac{3}{2})} - \frac{x^2}{2}\Big)\Bigg|_0^a\\ &= 2 \Big(\frac{2}{3}\sqrt{a}\cdot a^{\frac{3}{2}} - \frac{a^2}{2}\Big)\\ &= 2 \Big(\frac{2a^2}{3} - \frac{a^2}{2}\Big)\\ &= \frac{a^2}{3} \end{aligned}

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