Question: I'm a bit stuck on a geometric problem and after searching on Google, cannot find anything suitable - partly because I don't really know what to search for.

I've got a number of cross sections from along a river channel and want to work out a generalised volume for the channel. The cross sections are all separated by known distances, but the sizes and shapes vary. 

I am not very au fait with maths, but I believe the Cavalieri principle can be applied when the shapes vary but the area remains the same?.. or have I got this wrong and can it be used or adapted to work for this problem?

I'd be greatful if anybody could help me with a formula or point me in the right direction of where to look?

The image below illustrates what I'm talking about.

Thanks
Cobain

![A and B are different shapes with different known areas separated by a known distance](https://i.stack.imgur.com/WeSWN.png)

Answer: 
There are $9$ one digit numbers, which fill the first $9$ positions. There are $90$ two digit numbers, which fill the next $180$ positions. There are $900$ three digit numbers, which fill the next $2700$ positions. The position at the end of the $k$ digit numbers is $\sum\_{i=1}^k9i10^i$ You need to sum this series and find the smallest $k$ such that the sum is greater than $10^{100005}$
So, define $g(n)$ to be the number of digits of the number at the $n$th digit of this number.

You are looking for $f(n)=g(10^n)$.

$g(n)=k$ if and only if:

$$1+\sum\_{i=1}^{k-1} i(10^{i}-10^{i-1})\leq n < 1+\sum\_{i=1}^{k} i(10^i-10^{i-1})$$

So you need a closed form for $$m(k)=1+\sum\_{i=1}^{k} i(10^i-10^{i-1})=1+9\sum\_{i=1}^k i10^{i-1}$$

It can be shown that:

$$\sum\_{i=1}^{k-1} iz^{i-1} = \frac{(k-1)z^k - kz^{k-1}+1}{(z-1)^2}$$

With $z=10$ this gives $$m(k)=\frac{1}{9}\left((k-1)10^k-k10^{k-1}+10\right)$$

Then you are trying to find the smallest $k$ so that $m(k)>10^{100005}$.

Definitely $k\approx 100000$. You'd have to try some values.