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

Question

William Abdalla · Accepted Answer

Divide the area of the upper half of the ellipse by the length of the #x# axis of the ellipse to find that the average positive #y# coordinate is:

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

This ellipse passes through #(a, 0)#, #(0, b)#, #(-a, 0)# and #(0, -b)# The area of the ellipse is #pi a b# So the area of the upper half of the ellipse is #(pi a b)/2# The base of the upper half of the ellipse is of length #2a# So the average positive #y# value is #((pi a b) / 2)/(2a) = (pi b) / 4#

Isabella Ackerman · Answer

undefined 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:

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

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

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

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

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

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

$$ = \frac{b\pi}{2} $$

Isaiah Allen · Answer

undefined 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} $.