Volume of a cylinder, horizontally

[trancefishy]

i posted a question earlier, but, the heart of the matter has been narrowed down. this question is much more straightforward than the details in the other one. what i want to know is how to obtain the volume of a cylinder of radius r and height h by integration.
EASY. if you divide it into disks and add them up. no problem. i need the volume when it is divided into rectangles. as in, the cylinder is lying on it's side.

try and try i might, i always end up with either the square root of a negative number, or the entire integral turns out to equal zero.

also, even though i know the integral should equal pi r^2 h, i don't see how to get pi into the equation. the only way i can see to define the width of the rectangles is by using the pythagorean thereom on a general triangle on the sides of the cylinder. any insight into this would be greatly appreciated

 

[Tide]

Why don't you try something simpler first to get a handle on it? Try using rectangular coordinates to find the area of a circle.

 

[trancefishy]

hm, good thinking. i think I've done that before, but i will do it again. perhaps that will shed some light on this. this problem is just getting narrowed down thinner and thinner.

 

[ehild]

i need the volume when it is divided into rectangles. as in, the cylinder is lying on it's side.

try and try i might, i always end up with either the square root of a negative number, or the entire integral turns out to equal zero.

also, even though i know the integral should equal pi r^2 h, i don't see how to get pi into the equation. the only way i can see to define the width of the rectangles is by using the pythagorean thereom on a general triangle on the sides of the cylinder. any insight into this would be greatly appreciated

Show what you have done.

ehild

 

[trancefishy]

it took me a couple minutes to realize, this was the problem. i looked, and saw it was a simple trig substitution. i kid you not, i have spent, in the past 4 days, over 8 hours, solid, on and off, sometimes 2 hours at a time, wrestling with this problem. and it was a trig substitution the entire time that i was missing. that is why i couldn't integrate the cursed square root.

i feel like, overwhelmingly, stupid.

thank you very much for that advice, i am now on the right track. i wish i would have known this much much sooner.

thanks again