Why is option + s = ß and option + b = ∫?

A definitive answer would probably require being in the head of the person who defined those assignments. Yes, some option & shift + option characters do have shape relationships. Others have phonetic relationships. Option + s = ß is one of those, since ß as a sound character is related to s. I cannot even speculate on what the basis of option + b = ∫ is.

You would have to go way back in time to find out why. Here is the assignment published by Apple in the manual for the original Mac in 1984. (Note that the colors listed are wrong, Red should be White and Black should be Red.)

Those assignments have remained the same with a lot of additions. The following screenshots of the Keyboard Viewer show all of the current key assignments. The Keyboard Viewer is accessed with Input menu > Show Keyboard Viewer. If you don't have an Input menu (it is a flag icon representing your current input language), go to System Preference > Keyboard > Input Sources & check Show Input menu in menu bar.

Unmodified keys...

Option modified keys...

The keys with the orange surround are for accent marks. You type option + key, then an eligible character, so option + e followed by e gives é. If it is followed by an ineligible character, such as option + e followed by w, you get the accent followed by the character, like ´w.

Shift + option modified keys...

The why is impossible to answer without asking the person, who decided it so. However I would say that a big part of these key assignments is the simple fact that there are few keys on the keyboards, and lots of symbols you would like to produce - therefore you cannot assign a meaningful key to every symbol.

For the ß symbol the "s" key makes a great deal sense, as the symbol is really a form of s in for example German where it is used quite a lot. Here in Denmark the symbol is known as the "double-s". In German you would sometimes (especially in older texts) see it spelled out as "sz" if a ß symbol wasn't available.

For the ∫ symbol there's simply not a logical connection between the key b and the symbol itself - it would have been more logical that it would be matched with the s key, but as now know, it is already taken by the ß symbol. One interesting bit of information though is that the character code for the ∫ symbol is actually 222B, where as other forms of integral symbols have codes 222C, 222D, etc. I don't know if the Mac keyboard shortcut precedes the Unicode character code assignment or vice versa, but it could be a part of the thought process in assigning the seemingly random b key to that symbol.