## Calculate Volume of a Lemon-Use 3 methods

1. The problem statement, all variables and given/known data

Hi there, I have this project I need some help with. My group and I must calculate the volume of a lemon using three methods:

a) A Rotation of a Riemann Sum resulting in ‘discs’. Any type (left, right, midpoint, or trapezoid) is acceptable, although some are considerably more difficult.
b) Modeling the lemon’s shape using a function (or series of functions) and rotating it around the x-axis
c) Rotating a function around the y-axis using the cylinder method.

Here is a picture of our lemon

-For method a, we decided to do a trapezoidal sum.
-On method b, we are currently stuck on trying to model the lemon onto a graph.
-We have yet to start working on method c.

-Our main problem with this project is how to create an equation for our lemon.

-The lemon is 12.5 cm in length and 20.955 cm in width.

2. Relevant equations

Trapezoid sum equation:

-[b]= 16 [a]=0
Model Equation:
Work in Progress

Rotation along x and y axis:
Work in Progress

3. The attempt at a solution

-Using graphmatica and Wolfram 8 to model the Lemon. No success so far.

-Have yet to rotate lemon around x and y axis.

-Thanks for the help!
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 I haven't been using graphmatica, neither mathematica. But i probably would follow these steps if i were given such a nice looking lemon to model =) 1) Cut the lemon from center into 2 pieces. 2) Take one of the pieces and place it just above the x axis (i'd just hold it with hand while taking a picture) Note that the bottom of the piece must be precisely flat in order to see it is parallel to the x axis. 3) Roughly draw the contour line of the lemon above the x axis using a program like paint ( =) ) 4) I don't exactly how right now but i'd find a way to sample the points of the line so i get the numerical values of y versus x of the y(x) function of half lemon. 5) Use curve(polynomial probably) fitting to the data acquired from sampling. I would try different models to fit and choose the best one. 6) Find the coefficients for the model chosen. 7) Ta da! You got the function! Since you'll use rotating, no need to do the same for the other side of the lemon (although lemons are not so perfectly symmetrical beings) I'd probably use matlab for these operations (except for cutting the lemon of course). This method is the first one i came up with so it is probably the most harsh method for this problem of yours. Hope that it helps :)
 Hi, ckutlu. Thanks for the reply. :) My group and I have tried cutting the lemon and tracing it. From measurements we have taken, our lemon is 12.5 cm is in length. So, I'm writing the function as: S[integral] b=12.5 a=0 f(x) dx The problem I have is to determine the function of the lemon. I just cannot seem to come up with one that matches the shape of our lemon. Thanks again for the help! EDIT Really sorry, I had a small error in my measurements. The length of the lemon is 12.5 cm, and the width is 20.955 cm.

## Calculate Volume of a Lemon-Use 3 methods

 Quote by supahman Hi, ckutlu. Thanks for the reply. :) My group and I have tried cutting the lemon and tracing it. From measurements we have taken, our lemon is 12.5 cm is in length. So, I'm writing the function as: S[integral] b=12.5 a=0 f(x) dx The problem I have is to determine the function of the lemon. I just cannot seem to come up with one that matches the shape of our lemon. Thanks again for the help! EDIT Really sorry, I had a small error in my measurements. The length of the lemon is 12.5 cm, and the width is 20.955 cm.

Hmm i guess i couldn't explain what i meant. Ok, besides from the numerical approach i tried to explain in my previous post, you can just simply add few functions into another to find the most "lemony" looking one and scale them for the width and height you measured. Excuse me, that i can't offer anything other than an advice right now but hope you can find a way.

 Quote by ckutlu Hmm i guess i couldn't explain what i meant. Ok, besides from the numerical approach i tried to explain in my previous post, you can just simply add few functions into another to find the most "lemony" looking one and scale them for the width and height you measured. Excuse me, that i can't offer anything other than an advice right now but hope you can find a way.
Ah, okay that makes sense. I will try that out and report what I find.Thanks!
 Sorry for the bump, but this is to report what I've found so far. After splitting the equation into different parts, I got this equation. However, there seems to be a problem with the first integral. Could someone explain what looks wrong?
 In the first integral you put dx outside the ) of the integral.
 I think there is some deeper problem with that statement. I don't know mathematica well but looking at the expressions, i don't think there is something to plot. Since you are taking definite integrals all x will disappear and you will get only a constant referring to the area below the functions you used. Also what's up with the equal sign? I'll guess you are trying to use "some parts" of the functions in order to get the desired function but your approach is wrong. If you are going to use symbolic expressions then you should use heaviside functions to "bracket" the functions between the desired x values (you can look it up from google, mathematica probably has a built-in heaviside function). Or if you are going to use numerical approach, you should simply create a domain vector for x values. Say, between 0 and 50 with 0.01 increments. Then you create your functions using that x domain and add the values of the desired elements of one of the functions to the values of corresponding elements of the other function's vector. Tell me if it is not clear for you.
 Hi ckutlu, thanks for the reply. I fixed the equation a bit, but it still does not work. Could you explain heaviside a bit more?
 It is a function (H(x)) whose value is 1 for x>0 and 0 for x<0. It can be used to take "portions" of functions. You can google it for more details. I think i might be leading you to a dead end, i just gave few suggestions for you to try, not an exact answer. So if you think that trying these things will take more time than doing the actual work, try to find another approach for your problem. By the way is that a school project? If it is what grade?
 Thanks for all the help! I was able to solve this problem and finish the project today. :)
 Congrats! =)
 How would you go about doing a trapezoidal sum for this type of project. Here is our picture of the lemon: Attached Thumbnails

 Tags calculus, lemon, object, rotation, volume

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