Question

Asked Feb 20, 2019

Step 1

Please note that the volume of the ellipsoid is made up of aggregation of volume of various circular disc of radius y and thickness dx. One such disc is shown in the picutre on the white board.

Throughtout the answer to this question, we will make use of standard equation of ellipse that is x^{2} / a^{2} + y^{2} / b^{2} = 1 where length of major axis = 2a = 124 ft i.e. a = 124 / 2 = 62 and length of minor axis = 2b = 36 i.e. b = 36/2 = 18. We will do all the calulation using "a" and "b" substitute their values in the end.

Step 2

Consider the circular disc shown as hatched portion in the diagram above. It\'s radius = y. Hence, area, A = pi x (radius)^{2} = pi.y^{2}; its thickness dx. Hence, its volume, dV = A.dx = pi.y^{2}dx

From equation of ellipse, x^{2} / a^{2} + y^{2} / b^{2} = 1

Hence, y^{2} = b^{2} x (1 - x^{2} / a^{2})

Hence, volume of the disc, dV = pi.y^{2}dx = pi. b^{2} x (1 - x^{2} / a^{2}).dx

Range of integration will be x = -a to x = a. Since the shape is symmetrical, integration over x = - a to x= a will be equal to 2 x integration over x = 0 to x = a.

Step 3

Please see the white board f...

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