Volume Ellipse: Find x,a,b Rotation X-Axis

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SUMMARY

The volume of an ellipse defined by the equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) when rotated about the x-axis is calculated using the formula \(V = \pi ab^2\). Here, \(a\) represents the semi-major axis and \(b\) denotes the semi-minor axis. To determine the volume, one must first identify the value of \(x\) by setting \(y = 0\), resulting in \(x = a\). The final volume formula simplifies to \(V = \pi a^2 b^2\) after substituting the appropriate values.

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Find the volume of an ellipse [tex]\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1[/tex] after being rotated over the x-axis.
 
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Area generated by the solid fig will be given by

[tex]\int_{-a}^{a} \pi y^2 dx[/tex]
 


To find the volume of an ellipse after being rotated over the x-axis, we can use the formula V = πab^2, where a and b are the semi-major and semi-minor axes of the ellipse. In the given equation, a is the semi-major axis and b is the semi-minor axis.

First, we need to find the value of x when the ellipse is rotated over the x-axis. This can be done by setting y = 0 in the given equation, which gives us x = a.

Now, we can substitute the value of x in the formula V = πab^2 to get V = πa^2b^2. This is the volume of the ellipse after being rotated over the x-axis.

In summary, to find the volume of an ellipse \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 after being rotated over the x-axis, we use the formula V = πab^2, where a is the semi-major axis and b is the semi-minor axis. We also need to find the value of x when the ellipse is rotated, which can be done by setting y = 0 in the given equation.
 

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