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Question
A watermelon has an ellipsoid shape which can be obtained by revolving an ellipse with major-axis 20 cm and minor-axis 10 cm about its major-axis. Find its volume using integration
Solution
From the given data a = 10 cm and b = 5 cm
Equation of the Ellipse
`x^2/"a"^2 + y^2/"b"^2` = 1
`x^2/100 + y^2/25` = 1
`y^2/25 = 1 - x^2/100`
y2 = `25/100(100 - x^2)`
= `1/4(100 - x^2)`
Volume = `int_"a"^"b" piy^2 "d"x`
= `int_(-10)^10 pi/4(100 - x^2) "d"x`
= `(2pi)/4 int_0^10 (100 - x^2) "d"x`
= `pi/2[100x - x^3/3]_0^10`
= `pi/2[1000 - 1000/3]`
= `pi/2[2000/3]`
= `1000/3 pi "cm"^3`
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