# How to calculate the surface area of a normal chicken egg by using calculus? [duplicate]

I am recently doing my IB Maths HL internal assessment and my topic is how to calculate the surface area of an egg. I want to apply calculus knowledge into this question but my knowledge about this area is limited. I only know the basic information about calculus, for example, using integration to calculate the volume of an object. I have already searched for relevant information online for many days but I still have no idea. I need to finish my internal assessment in 2 weeks! Could anyone please help me with it?! Thank you!

Well, I know there has already been a same question but i don't really understand the answer for that question. I know it is related to integral but i have no idea how to do that. I hope someone can explain it in details. If it is just an idea of how to do it, I can search for ideas everywhere by using Google and have no need to ask for help here. I really need help like step by step maybe. Thanks.

[I have worked out the formula A=∫_(x_2)^(x_1)▒〖2πf(x)·√(1+f^' (x)^2 )·dx〗, however, f'(x) involves a constant C which is unknown. I am confused how can I solve the equation when I have no idea what C is.

## marked as duplicate by Adam Hughes, user99914, John B, Watson, Claude Leibovici calculus StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Mar 29 '16 at 9:16

• The relevant phrase here is probably "surface of rotation". – ChocolateAndCheese Mar 29 '16 at 6:24
• And the relevant question: what's the (mathematical) shape of an egg? – Jean-Claude Arbaut Mar 29 '16 at 6:29
• You could probably find two parabolas describing the top and bottom halves of the egg ("top"=narrow part, "bottom"=wider part), and then find the surface areas of those and add up. – A.Sh Mar 29 '16 at 6:30
• Better yet: osaka-ue.ac.jp/zemi/nishiyama/math2010/egg.pdf By Figure 1, you could try and fit two oval halves to the top and bottom parts of the egg, and proceed from there. Just measure the top-to-bottom and the side-to-side lengths of the egg, as well as where the bottom half switches into the top half, and boom. – A.Sh Mar 29 '16 at 6:34
• And in the related section: math.stackexchange.com/questions/407310/… – A.Sh Mar 29 '16 at 6:35

You need to start by thinking about what assumptions you can make about the egg's shape to simplify the mathematics.

Firstly - is the cross-section perfectly circular? This would be a good assumption to make the mathematics significantly easier (appropriate for IBDP HL)

Secondly - what is the side profile of an egg? Maybe you have already researched this, I don't know. If not boil an egg, cut it in half long ways and trace out the shape. You have hopefully done some function study and modeling in IBDP HL Math so can approximate its shape with function(s). Including all these aspects in your IA would be good as they help you demonstrate a good understanding of the various parts of the course and how they can link together.

Lastly - once you have function(s) $f(x)$ describing its shape you can find the surface area of revolution with the following formula:

$$S = 2\pi\int_a^b f(x)\sqrt{1+\left(f'(x)\right)^2} \,dx$$

Depending on the functions you have chosen and your level of confidence with integration you may not be able to complete the integral. That would then require you to go back and try to approximate the shape of the egg with simpler functions.

• Hi, thank you for your answers but i didn't get how do you get the formula S=2π∫ balabala...? And how can i get the function of the side profile of the egg? – Tarina Xu Mar 31 '16 at 8:15
• The formula is the formula for the surface area of revolution. That is the Mathematical Exploration aspect of your IA. You should be doing further reading beyond what you do in class to find solutions. As for finding the function of the side profile look back at all the functions you learnt in Topic 2 - Functions and equations - something there will be a close match to the shape of an egg. Finally remember that an IA is meant to be your own work so we can give advice and assistance but you can't get others to do it for you. – Ian Miller Mar 31 '16 at 14:38