# Negative Volume of Revolution?

There is a nice equation made by Nobuo Yamamoto which describes the curve of an egg and it is:

(x^2 + y^2)^2 = ax^3 + (3/10)xy^2, where a is the length of the major axis of the egg.

Solve this equation for y, we get:

y=+/- sqrt((3/20)ax - x^2 + xsqrt((7/10)ax + (9/400)a^2))

When I rotate the function around the x-axis by 2(pi), the result is a negative volume of (-1)((pi)(a^3))/12.

I don't know what I am doing wrong, or how I can fix this problem.

phinds
Gold Member
2021 Award
Show your work. How else might we see where you went wrong?

Show your work. How else might we see where you went wrong?
Sorry, new to this site.

I am using the positive case of the equation mentioned.     vela
Staff Emeritus
Homework Helper
It looks like you didn't solve for ##y## correctly. I get
$$y = \sqrt{\left(\frac b2\right)x-x^2+x\sqrt{\left(\frac b2\right)^2+(a-b)x}}$$ where ##b=3/10##.

To integrate the term of the form ##x\sqrt{c+dx}##, try using the substitution ##u=c+dx##.

vela
Staff Emeritus