Draw the shape of a horizontally laying egg. Use a sphere and an ellipse to draw the egg, making sure they overlap in their vertical axis. Half of the ellipse is the left side of the egg, and half of the sphere is the right side.
Draw a vertical line dividing the egg in two nonequal parts. The vertical line must coincide with the minor vertical axis of the ellipse. Draw a horizontal line that divides the egg in two equal parts. The vertical line and the horizontal are your XY axis.
Label the points where your drawing intersects your XY axis. The point where both axes cross is the (0,0) point. The points in the vertical line are, from top to bottom: (0,b) and (0,-b).
The points on the horizontal line are, from left to right: (-a,0) and (0,b).
In our egg, b+b is the height of the egg, and a+b is the length.
Divide your drawing in two. On one half, keep the left part, with the ellipse. The other half keeps the part with the circle. Erase anything below the horizontal axis on both drawings. In the end, you should have two drawings that resemble the top left quarter of an ellipse, and the top right quarter of a circle.
Find the area of the circle. Use the volume by revolution formula. This formula rotates the quarter circle along the X axis to create a volume.
This is the equation of the volume by revolution:
Integrate the expression " Pi x ( b^2 - X^2 ) " from [0 to b].
Where:
Pi = 3.141592... Constant of the circle
(b^2 - X^2) = Equation of the circle squared
"^2" means "to the power of two"
[0 to b] means the bound for our integral, which is the point in the X axis in which our circle is drawn.
Solve the circle integral.
Factorize Pi:
Pi x [ integral( b^2 - X^2 ) ] from [0 to b]
Use the Online Integrator to solve the integral.
Pi x [ ( b^2 x X ) - ( X^3/3) from [0,b] ]
Replace 0 and b:
Pi x [ (( b^2 x b ) - ( b^3/3 )) - ( ( b^2 x 0 ) - ( 0^3/3 ))]
The answer is: ( 2 / 3 ) x Pi x b^3
Calculate the volume by revolution of the ellipse. The ellipse extends from [-a to 0] along the X axis. These points will serve as the boundaries of our integration.
This is the formula:
Integrate: "Pi x ( ( b^2 / a^2 ) x ( a^2 - X^2 ) )" from [-a to 0]
Where:
Pi = 3.141592... Constant of the circle
( ( b^2 / a^2 ) x ( a^2 - X^2 ) ) Equation of the ellipse squared
"^2" means "to the power of two"
Solve the ellipse integral. Factorize Pi:
"Pi x integrate( ( b^2 / a^2 ) x ( a^2 - X^2 ) )" from [-a to 0]
Use the Online Integrator to solve the integral (see References).
Pi x [ ( 1 / 3 )( b^2 )( X )( 3 - ( X^2 / a^2 )] from [-a to 0]
Replace -a and 0:
Pi [ ( ( 1 / 3 )( b^2 )( 0 )( 3 - ( 0^2 / a^2 ) ) - ( ( 1 / 3 )( b^2 )( -a )( 3 - ( ( -a^2 ) / a^2 ) ) ]
After simplifying the answer is:
( 2 / 3 ) x Pi x b^2 x a
Add the volume of the circle and the volume of the sphere. This is the total volume of the egg.
After simplifying, the answer is:
( 2 / 3 ) x Pi x b^2 x ( a + b)
Replace numbers for a and b. A large egg height is 2 inches, and length is 3 inches.
From the example:
height = b + b = 2 inches
b = 1 inch
length = a + b = 3 inches
a + 1 inch = 3 inches
a = 2 inches
The answer is:
( 2 / 3 ) x Pi x b^2 x ( a + b)
Replacing a and b:
( 2 / 3 ) x Pi x (1)^2 x ( 2 + 1 ) = 2 x Pi = 6.2831 cubic inches.