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):
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).
Since `a = 14` and `b = 12.5`, we have:
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`
`=2.504 times 1829.33`
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.
Easy to understand math videos: