# Negative Volume of Revolution?

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1. Jan 10, 2015

### sam9734

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.

2. Jan 10, 2015

### phinds

Show your work. How else might we see where you went wrong?

3. Jan 10, 2015

### sam9734

Sorry, new to this site.

I am using the positive case of the equation mentioned.

4. Jan 12, 2015

### vela

Staff Emeritus
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 $u=c+dx$.

5. Jan 12, 2015

### sam9734

6. Jan 13, 2015

### vela

Staff Emeritus
OK, so you have a typo in the equation in your first post. Your expression for $y$ is correct, but you didn't evaluate the third integral correctly.