I'm a high school student trying to model an egg-shaped object. I have found this formula describing my object: $$\frac{x^2}{10+3x}+\frac{y^2}{4}=0.5$$
The next step is to calculate the volume and surface area, and the only method I know is to integrate it. However, the equation is both over and under x-axis, so I can't use the solid of revolution formula. If I rotate it for only 180 degrees like this:
$$\int_0^1 \frac{\pi}{2}y^2 \;\mathrm{dx}$$
will it give me the volume of the thing? If it doesn't work like that, what are the methods I could use?
If the given equation describes the $xy$-plane cross-section of a solid of revolution whose axis of symmetry is the $x$-axis, then the equation for the surface will be given by $$\frac{x^2}{10+3x} + \frac{y^2 + z^2}{4} = \frac{1}{2}. \tag{1}$$
The volume of interest may be computed via the method of disks:
$$V = \int_{x=(3-\sqrt{89})/4}^{(3+\sqrt{89})/4} \pi \frac{20 + 6x - 4x^2}{10 + 3x} \, dx. \tag{2}$$
I leave the computation of the antiderivative and the subsequent volume as an exercise for the reader; it is not difficult.
The surface area is given by
$$S = 2\pi \int_{x=(3-\sqrt{89})/4}^{(3+\sqrt{89})/4} f(x) \sqrt{1 + (f'(x))^2} \, dx, \\ f(x) = \sqrt{\frac{20+6x-4x^2}{10+3x}}, \tag{3}$$ which is understandably quite messy to write out explicitly and does not admit an elementary closed form solution. Numeric integration is feasible, however.