It's All Conics!

For the conic $ax^2 + bxy + cy^2 + dx + ey = -16$ .

For $(0,4):\implies 4c + e = -4$

For $(0,-4):\implies 4c - e = -4$

For $(2,0):\implies 2a + d = -8$

For $(-2,0):\implies 2a - d = -8$

For $(1,4):\implies a + 4b + 16c + d + 4e = -16$

$\implies c = -1, e = 0, a = -4, d = 0$ and $b = 1 \implies -4x^2 + xy - y^2 = -16$ or $4x^2 - xy + y^2 = 16$ which is an ellipse since $b^2 - 4ac > 0$ .

Solving for $y$ we obtain:

$y = \dfrac{1}{2}(\pm\sqrt{64 - 15x^2} + x)$

Using the portion of the ellipse above the line $y = \dfrac{x}{2} \implies \boxed{y_{1}(x) = \dfrac{1}{2}(\sqrt{64 - 15x^2} + x)}$ .

For circle $(x - a)^2 + (y - b)^2 = r^2$ .

For (1) $(0,4):\implies a^2 + 16 - 8b + b^2 = r^2$

For (2) $(0,5):\implies a^2 + 25 - 10b + b^2 = r^2$

Subtracting (1) from (2) we obtain: $9 - 2b = 0 \implies b = \dfrac{9}{2}$ .

For (3) $(-1,3) \implies 1 + 2a + a^2 + 9 - 6b + b^2 = r^2$

For (1) $(0,4):\implies a^2 + 16 - 8b + b^2 = r^2$

Subtracting (1) from (2) we obtain: $a + b = 3$ and using $b = \dfrac{9}{2} \implies a = -\dfrac{3}{2}$

$\implies r^2 = \dfrac{5}{2} \implies (x + \dfrac{3}{2})^2 + (y - \dfrac{9}{2})^2 = \dfrac{5}{2}$ .

Solving for $y$ we obtain:

$y = \pm\sqrt{\dfrac{5}{2} - (x + \dfrac{3}{2})^2} + \dfrac{9}{2}$

Using the portion of the circle below the line $y = \dfrac{9}{2} \implies \boxed{y_{2}(x) = -\sqrt{\dfrac{5}{2} - (x + \dfrac{3}{2})^2} + \dfrac{9}{2}}$

You can check that $(-1,3)$ satisfies the ellipse above.

From graph above the ellipse and the circle intersect at $(-1,3), (0,4)$ .

For $\displaystyle\int_{-1}^{0} y_{1}(x) dx$ :

Let $I_{1} = \displaystyle\dfrac{1}{2}\int_{-1}^{0} \sqrt{64 - 15x^2} dx$ .

Let $\sqrt{15}x = 8\sin(\theta) \implies dx = \dfrac{8}{\sqrt{15}}\cos(\theta) d\theta \implies$

$\dfrac{32}{\sqrt{15}}\displaystyle\int \cos^2(\theta) d\theta = \dfrac{16}{\sqrt{15}}\displaystyle\int (1 + \cos(2\theta)) d\theta = \dfrac{16}{\sqrt{15}}(\theta + \dfrac{1}{2}\sin(2\theta)) =$ $\dfrac{16}{\sqrt{15}}(\arcsin(\dfrac{\sqrt{15}x}{8}) + \dfrac{\sqrt{15}x\sqrt{64 - 15x^2}}{64})$

$\implies I_{1}(x)|_{-1}^{0} = \dfrac{16}{\sqrt{15}}(\arcsin(\dfrac{\sqrt{15}}{8}) + \dfrac{7\sqrt{15}}{64})$

$\implies \displaystyle\int_{-1}^{0} y_{1}(x) dx = \dfrac{16}{\sqrt{15}}\arcsin(\dfrac{\sqrt{15}}{8}) + \dfrac{7}{4} - \dfrac{1}{4} = \dfrac{16}{\sqrt{15}}\arcsin(\dfrac{\sqrt{15}}{8}) + \dfrac{3}{2} \approx = \boxed{3.587736}$

For $\displaystyle\int_{-1}^{0} y_{2}(x) dx$ :

Let $I_{2} = -\displaystyle\int_{-1}^{0} \sqrt{\dfrac{5}{2} - (x + \dfrac{3}{2})^2} dx$

Let $x + \dfrac{3}{2} = \sqrt{\dfrac{5}{2}}\sin(\theta) \implies dx = \sqrt{\dfrac{5}{2}}\cos(\theta) d\theta$

$-\dfrac{5}{2}\displaystyle\int \cos^2(\theta) d\theta = -\dfrac{5}{4}\displaystyle\int (1 + \cos(2\theta)) d\theta = -\dfrac{5}{4}(\theta + \dfrac{1}{2}\sin(2\theta)) = -\dfrac{5}{4}(\arcsin(\dfrac{x + \dfrac{3}{2}} {\sqrt{\dfrac{5}{2}}}) + \dfrac{2}{5}(x + \dfrac{3}{2})(\sqrt{\dfrac{5}{2} - (x + \dfrac{3}{2})^2}))$

$\implies I_{2}(x)|_{-1}^{0} = -\dfrac{5}{4}(\arcsin(\dfrac{3}{\sqrt{10}}) - \arcsin(\dfrac{1}{\sqrt{10}}))$

$\implies \displaystyle\int_{-1}^{0} y_{2}(x) dx = -\dfrac{5}{4}(\arcsin(\dfrac{3}{\sqrt{10}}) - \arcsin(\dfrac{1}{\sqrt{10}})) + \dfrac{9}{2} \approx \boxed{3.340881}$

$\implies \displaystyle\int_{-1}^{0} y_{1}(x) - y_{2}(x) dx = \boxed{0.246855}$ .