I'm trying to solve a problem my class was given for homework - wherein we have to find the volume of fluid occupying a horizontal cylindrical tank at 100%, 90% and 80% capacity.

Since the hemispherical end caps of the tank are segments of a sphere, I reasoned that I'd have to try and build an expression to integrate with respect to the height of the tank.

The issue is, it's been a while since I've learned integrals, especially problems where I have to triple integrate and take weird boundaries.

I've also been given limited information, as I only have the

• straight length (6000mm)
• length running down the segment radius of the tank (2100mm)
• depth of the segment (x = 145mm)

My attempts so far:

Attempt #1: Try and 'brute force' it - by ignoring the end caps initially, finding the volume of the central cylinder at 100% full. Based on the height of the fluid, I tried working with the ratio of full/empty space in the tank and attempted to use simple arithmetic to work out the remaining volume within the tank occupied by fluid. This effort failed.

Attempt #2: I tried triple integrating with respect to cylindrical coordinates (x,y,z) but got very confused as to how to find the radius of the sphere the segment is a part of.

Attempt #3: Followed some YouTube videos and tried to integrate with spherical coordinates (rho,theta,psi) but got confused again - as I was unsure of how to take my boundaries, since I'm unsure as to if I'm supposed to take my reference point as the centre of the sphere or shift it downward. And after that point I'm just not sure how to proceed.

Any help or advice would be really appreciated.

Thanks in advanced!! :)

idk if this is the right place; this is more general math class advice rather than a specific problem.

I need to get through pre-calc, calc 1, and calc 2 for my degree. The problem is I im in pre-calc/trig, and I feel like I'm just stuck. My big issue is formulas. I could stare at the trig unit circle for a year straight and barely remember when I look away. Thankfully, I found the finger trick for that one, but I still feel stuck on how I'm supposed to get through Calc 1 and 2 when im barely floating in trig/precalc. Does anyone have advice for not being able to memorize formulas, or just advice in general? I feel like I'm up against a brick wall.

I don't really know how to show proof of my prior attempts for rule 2.

I've tried brute force memorization, and it never works. I find that if I am about to have an anxiety attack about learning, I learn the stuff better, but I would rather avoid that., some times tricks work, but only if paired with already being stressed abt it

So I’m writing this math paper for school. I solved it graphically but I really wanted to do it by hand so that the work could be shown, cuz it’s like extra swag yk? Anyways, I reached out to a teacher about it and she was like “I don’t think it’s possible to solve it algebraically” and I was like “I don’t think that’s possible” anyways it goes like this 145.04714x(0.991479)^(x-1)>-1 
Then solving it for x.
If anyone wants to read the paper and give feedback that would be very helpful. (Please if you are like transphobic or something, please just look at the math, this girl in my class is transphobic and it was kinda just a waste of time because her feedback was so obviously biased)
I already know that I don’t have the axis labeled btw.