# How do you find the average value of the positive y-coordinates of the ellipse #x^2/a^2 + y^2/b^2 = 1#?

Divide the area of the upper half of the ellipse by the length of the

#((pi ab) / 2) / (2a) = (pi b) / 4#

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To find the average value of the positive y-coordinates of the ellipse ( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 ), you integrate the y-coordinate over the interval where y is positive and then divide by the length of that interval. The equation of the ellipse implies that ( y = b\sqrt{1 - \frac{x^2}{a^2}} ) when ( x ) is in the interval ([-a, a]). Integrating ( y ) over this interval and dividing by the length of the interval ( 2a ) gives the average value of the positive y-coordinates:

[ \text{Average} = \frac{1}{2a} \int_{-a}^{a} b\sqrt{1 - \frac{x^2}{a^2}} , dx ]

[ = \frac{b}{a} \int_{-a}^{a} \sqrt{1 - \frac{x^2}{a^2}} , dx ]

[ = \frac{b}{a} \left[\frac{a^2}{2}\sin^{-1}\left(\frac{x}{a}\right)\right]_{-a}^{a} ]

[ = \frac{b}{a} \left[\frac{a^2}{2}\left(\sin^{-1}(1) - \sin^{-1}(-1)\right)\right] ]

[ = \frac{b}{a} \left[\frac{a^2}{2}\left(\frac{\pi}{2} - \left(-\frac{\pi}{2}\right)\right)\right] ]

[ = \frac{b}{a} \left[\frac{a^2}{2}\pi\right] ]

[ = \frac{b\pi}{2} ]

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To find the average value of the positive y-coordinates of the ellipse ( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 ), integrate ( y ) over the interval where ( y ) is positive and divide by the length of that interval. The interval for positive ( y ) values on the upper half of the ellipse is from 0 to ( b ). Thus, the average value of the positive y-coordinates can be found using the formula:

[ \text{Average} = \frac{1}{b} \int_{0}^{b} y , dy ]

Integrate ( y ) with respect to ( y ) from 0 to ( b ), and then divide the result by ( b ).

[ \text{Average} = \frac{1}{b} \left[ \frac{y^2}{2} \right]_{0}^{b} ]

[ = \frac{1}{b} \left( \frac{b^2}{2} - \frac{0^2}{2} \right) ]

[ = \frac{1}{b} \left( \frac{b^2}{2} \right) ]

[ = \frac{b}{2} ]

So, the average value of the positive y-coordinates of the ellipse ( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 ) is ( \frac{b}{2} ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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