Can the Volume of Revolution Be Negative?

[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.

Can someone please help me out on this?

 

[phinds]

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

 

[sam9734]

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]

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 $u=c+dx$.

 

[sam9734]

I used the following site for the equation mentioned.

http://www.geocities.jp/nyjp07/Egg/index_egg_E.html

I used equation 9b and substituted b = 0.7a into it.

 

[vela]

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.