We are told the melon is an ellipsoid. We need to find the equation of the cross-sectional ellipse with major axis 28 cm and minor axis 25 cm.

We use the formula (from the section on ellipses):

`(x^2)/(a^2)+(y^2)/(b^2)=1`

where a is half the length of the major axis and b is half the length of the minor axis.

For the volume formula, we will need the expression for y2 and it is easier to solve for this now (before substituting our a and b).

`(x^2)/(a^2)+(y^2)/(b^2)=1`

`b^2x^2+a^2y^2=a^2b^2`

`a^2y^2=a^2b^2-b^2x^2=b^2(a^2-x^2)`

`y^2=(b^2)/(a^2)(a^2-x^2)`

Since `a = 14` and `b = 12.5`, we have:

`y^2=(12.5^2)/(14^2)(14^2-x^2)`

`=0.797(196-x^2)`

NOTE: The a and b that we are using for the ellipse formula are not the same a and b we use in the integration step. They are completely different parts of the problem.

Using this, we can now find the volume using integration. (Once again we find the volume for half and then double it at the end).

`text[V]_[text[half]]=pi int_0^14y^2 dx`

`=pi int_0^14 0.797(196-x^2)dx`

`=0.797pi int_0^14(196-x^2)dx`

`=2.504[196x-(x^3)/(3)]_0^14`

`=2.504[196(14)-(14^3)/(3)]`

`=2.504 times 1829.33`

`=4580.65\ text[cm]^3`

So the watermelon's total volume is `2 × 4580.65 = 9161\ "cm"^3` or `9.161\ "L"`. This is about the same as what we got by slicing the watermelon and adding the volume of the slices.

Get the Daily Math Tweet!