Question

An ellipse is rotated along x axis find the volume genrated by curve

Answer 1

Answer:

If the ellipse is given by

( x² / a² ) + ( y² / b² ) = 1

then the volume is

V = 4 π a b² / 3.

Step-by-step explanation:

The values of x range from -a to a.

At any particular x value, taking a "slice" of the volume of width dx, the volume of this slice is dV = π y² dx, since it is a disk with radius y and thickness dx.  From the equation for the ellipse, this can be put in terms of x:

dV = π b² ( 1 - x² / a² ) dx.

To get the volume of the solid, we "add together" these slices, so

V = ∫ dV  (integrate from x = -a to x = a)

= 2 ∫ dV  (integrate from x = 0 to x = a, using symmetry to make things tidy)

= 2 π b² ∫ ( 1 - x² / a² ) dx  (integrate from x = 0 to x = a)

= 2 π b² [ x - x³ / ( 3 a² ) ]    (evaluate from x = 0 to x = a)

= 2 π b² ( a - a³ / ( 3 a² ) )

= 2 π b² ( a - a / 3 )

= 2 π b² ( 2a / 3 )

= 4 π a b² / 3