# average height of a point on an arc vs hemisphere

Why isn't the average height of a point on an arc of radius a the same as the average height on a surface of radius a.

Stated another way the first problem is: Find the average height of a point on a unit semicircle with respect to the arc length θ.

The second is: Find the average height above the xy-plane of a point chosen at random on the surface of the hemisphere $x^2 + y^2 + z^2 = a^2$.

I thought both questions should have the same answer but.

For arc $$\bar{y} = \frac{1}{\pi}\int_0^\pi a\sin\theta \ \mathrm{d}\theta = \frac{2a}{\pi}$$

For surface

\begin{align} \bar{z} &= \frac{\int\int_S z\ \mathrm{d}S}{\int\int_S \mathrm{d}S}\\ &= \frac{\int_0^{2\pi}\int_0^{\pi/2}a\cos\phi\ a^2\sin\phi\ \mathrm{d}\phi\ \mathrm{d}\theta}{\int_0^{2\pi}\int_0^{\pi/2}a^2\sin\phi\ \mathrm{d}\phi\ \mathrm{d}\theta} \\ &= \frac{a}{2} \end{align}

I'v posted a link which covers what I mean by average height for an arc as well as problem set(sol) where I originally encountered the half-sphere problem. Check it out as they might explain the problems better.

average-height-on-arc

look for question 6B-12

half-hemisphere