( a x ) 2 + ( b y ) 2 = 1
An ellipse is described by the equation above. What is ithe volume of the solid that is obtained by rotating the ellipse around the x -axis?
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The ellipse intersects with the x-axis in P ( a ; 0 ) and Q ( − a ; 0 ) , which can be obtained from the above equation for y = 0 . We have ( a x ) 2 + ( b y ) 2 = 1 , which is equivalent to y = ± b 1 − a 2 x 2 . For the part of the ellipse that lies above the x-axis the sign will be + . Now let f ( x ) = b 1 − a 2 x 2 , which implies [ f ( x ) ] 2 = b 2 ( 1 − a 2 x 2 ) . Hence by the application of the disk method , the volume is:
V = π ∫ − a a ( b 2 − a 2 x 2 ) d x
V = 2 π ∫ 0 a ( b 2 − a 2 x 2 ) d x
V = 2 π [ b 2 x + 3 a 2 b 2 x 3 ] 0 a
V = 2 π ( b 2 a + 3 a 2 b 2 a 3 )
V = 3 4 π a b 2