$\text{Q Find the area enclosed by the curves "}y^2+x^2=9\text{" and "}\left|\left(x^2-y\left|x\right|\right)\right|=1$

Question

$\text{Q Find the area enclosed by the curves "}y^2+x^2=9\text{" and "}\left|\left(x^2-y\left|x\right|\right)\right|=1\text{" which contains the origin}.$

I tried to plot the graph on desmos. and got the following graph--

I am unable to approach how to get the area. perhaps definite integration would work, but I am not able to integrate it. Direct integration of the function is not possible, hence breaking it into parts would be required. the graph is symmetric on the y-axis, hence if only half the area is found out, we can deduce the next. I am also unfortunately unable to figure out what the limits would be,(perhaps I guess they need to be obtained by solving the circle and the second curve.) Also, there is an issue that a circle cannot be integrated directly (as it is not a function). I think to subtract the small areas from the total area of the circle would be a better idea. please help. Thanks in advanced.

Answer 1

Hint: you can split $\left|\left(x^2-y\left|x\right|\right)\right|$ into four functions, then, split the area you are trying to find into manageable pieces each one with a well defined function on top and bottom.

Answer 2

\begin{align} f_1(x)&=\phantom{-}\sqrt{9-x^2}\quad\text{(yellow)} ,\\ f_2(x)&=-\sqrt{9-x^2}\quad\text{(red)} ,\\ f_3(x)&=\frac{x^2-1}x\quad\text{(orange)} ,\\ f_4(x)&=\frac{x^2+1}x\quad\text{(blue)} . \end{align}

\begin{align} A&=(x_{14a},f_1(x_{14a})) = \left(\tfrac12\,\sqrt {7-\sqrt {41}}, \tfrac12\,\sqrt {29+\sqrt {41}}\right) \approx(.3862886754,2.975026228) ,\\ B&=(x_{14b},f_1(x_{14b})) = \left(\tfrac12\,\sqrt {7+\sqrt {41}},\tfrac12\,\sqrt {29-\sqrt {41}}\right) \approx (1.830513878,2.376808562) ,\\ C&=(x_{13},f_1(x_{13})) =\left( \tfrac12\,\sqrt {11+\sqrt {113}},\tfrac12\,\sqrt {25-\sqrt {113}} \right) \approx (2.325411028,1.895379526) ,\\ D&=(x_{23},f_1(x_{23})) =\left( \tfrac12\,\sqrt {11-\sqrt {113}}, -\tfrac12\,\sqrt {25+\sqrt {113}} \right) \approx (.3040781930,-2.984549623) . \end{align}

\begin{align} S_1&=\int_{A_x}^{B_x} f_1(x)-f_4(x)\, dx \approx 0.8164322093 ,\\ S_2&=\int_{D_x}^{1} f_3(x)-f_2(x)\, dx \approx 1.296107328 ,\\ S_3&=\int_1^3 -f_2(x)\, dx \approx 4.125103818 ,\\ S_4&=\int_1^{C_x} f_3(x)\, dx \approx 1.359871417 ,\\ S_5&=\int_{C_x}^3 f_1(x)\, dx \approx 0.873621585 ,\\ \sum_{i=1}^5 S_i&\approx 8.47113635 . \end{align}