I'm slightly confused about how it works with regards to the following out of my textbook;
"If we can consider a solid to be made up of lots of thin discs stacked on top of each other, and then find an expression for the cross-sectional area of each disc, then the volume of each disc is $\delta V=A(h)\times \delta h$, where $A(h)$ is the cross-sectional area at a height $h$ and $\delta h$ is the thickness of the disc"
(I understand all of this fine)
"The total volume of the solid can be found by adding the volumes of all the discs; $V=$ sum of all $\delta V =$ sum of all $A(h) \times > \delta h$"
(I understand this)
$$\therefore V=\lim_{\delta h \to0} \sum_{x=a}^b \delta V= \lim_{\delta h \to0} \sum_{x=a}^b A(h) \times \delta h$$
This process can be represented by $V=\int_a^b A(h)dh$
I don't understand this? How does the finite sum $V=\lim_{\delta h \to0} \sum_{x=a}^b \delta V$, become an integral of an infinite sum?
Because the summation notation the way they have written it, it indicates a sum of only $(b-a)$ times. Doesn't it?
In another textbook I understand how they write it as;
$$\sum_{i=1}^n \Delta V_i = \sum_{i=1}^n \pi [f(x_i)]^2 \Delta x_i$$
Because, this can be seen as a Riemann Sum, so as $\lim_{n \to \infty} \sum_{i=1}^n \Delta V_i = \pi \int_{a}^b [f(x)]^2 dx$
But I don't understand the first textbooks notation?
Thanks
I'm going to write out what I think this notation means, which is that we divide $[a,b]$ into $n$ equal intervals of length $\delta h$. Then $\delta h = (b-a)/n$, which I hope is obvious. Then the sum is $$ \sum_{k=0}^n A(a+k \delta h) \, \delta h: $$ the top end is then $a+n\delta h = a+b-a=b$. One could then write this as $$ \sum_{h=a,a+\delta h, \dotsc}^b A(h) \, \delta h, $$ which is a bit clearer than your book's notation, but still pretty awful.
But you are right to complain: summation notation is normally used exclusively with an index that increments in integer steps, and there is no indication in that notation that the increments are actually $\delta h$ instead of $1$. Presumably it is done as some sort of sneaky way of making it look like an integral without actually going into the details of the limit, but instead it's just unnecessarily confusing: it doesn't specify the increment of the index, nor what the argument of the function is properly, so you actually can't tell what the sum means except by the immediate context.
I think you should write to the author; I suspect you're neither the first nor the last person to object to this.