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

[tandoorichicken]

Find the volume of an ellipse \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 after being rotated over the x-axis.

 

[himanshu121]

Area generated by the solid fig will be given by

\int_{-a}^{a} \pi y^2 dx

 

[M98Ranger]

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.