Problem Solving > How to calculate the volume of an egg using integral calculus

Although this is a step by step guide to how to find the volume of an egg using calculus, you should be familiar with basic integration before working through the steps.

## How to Find the Volume of an Egg using Calculus: Steps

Step 1: **Draw the egg using an ellipse and sphere** and make sure that they overlap each other. Split the egg into two parts by drawing a line over the vertical axis of the ellipse and draw another horizontal line that splits the egg into two equal parts. These are your X and Y axes.

Step 2: Name the points where the egg intersects the two axis as follows: (0, n ) and (0, -n ) on the vertical line and (-m, 0) and (m, 0) on the horizontal one.

Step 3: **Split your egg into 2 parts.** Have the left part of the ellipse in one half and the circle in the other. Remove anything below the horizontal line and you should now have two drawings, one with the upper left part of an ellipse and the other with the upper right part of the circle.

Step 4: **Calculate the circle’s area** and use the **volume by revolution formula **which rotates the circle along the X axis resulting in volume.

This is the equation: Integrate π (n^{2}-X^{2}) from 0 to n.

You can use this online integrator to help you.

Step 5 : **Calculate the circle integral **and factorize π:

π [∫(n^{2}-X^{2})] from [0 to n]

Solve the integral by using the online integrator.

Replace 0 and n and you should get

π [((n^{2} * n) – (n^{3/3})) – ((n^{2} * 0) – (0^{3/3}))]

resulting ( 2/3 ) π * n^{3}.

Step 6: **Find the volume of the ellipse.** Do this by integrating π( (n^{2}/m^{2}) (m^{2}-X^{2}) ) from [-m to 0 ].

Step 7: **Solve the integral of the ellipse**, and factorize π:

π * ∫ ((n^{2}/m^{2})*(m^{2}-X^{2})) from [-m to 0]

Use the Online Integrator again.

You will get:

π * [(1/3)(n^{2})(X)(3-(X^{2}/m^{2})] from [-m to 0]

After replacing -m and 0 and doing the simplifications your result will be (2/3) π * n^{2} * m.

Step 8: **Get the total volume of the egg **by summing up half the volume of the sphere with half the volume of the ellipse. After simplification the answer is (2/3) π * n^{2} * (m+n).

Step 9: **Replace m and n with numbers** in the (2/3) π * n^{2} * (m+n) equation and your result should be around 1.8 cubic inches, more or less depending on the size of the egg.

**Here is an example:**

An average chicken egg is 5.6 cm long and has a diameter of 4.4 cm at its widest part. Now let’s replace m and n with the proper numbers in the last formula.

Diameter is 4.4 so n= 2.2 cm

Length is m + 2.2 = 5.6 so m= 3.4

(2/3) π * 2.2^{2} * (2.2 + 3.4 ) = 28,368 cubic centimeters which is almost 2 cubic inches, 1.73 to be more precise.

*That’s How to Find the Volume of an Egg !*

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