Mixing
Part 1 - Trying to Derive General Formula for Volume in Partially Filled Oblique Cylinder
$begingroup$
I am trying to find a general formula for the volume in a partially filled oblique cylinder (specifically using "dx slices" as the area of the segments). I have set up an integral, which I WAS ABLE to solve numerically and produce correct numerical results, as checked against: https://planetcalc.com/1442/ (their equation set up differently than mine).
So that gives me the ??assurance?? that my integral is a legitimate one. But when I actually complete the integration, it does not produce the correct results. Seeing as I now have a satisfactory solution due to my numerical solving, this exercise is no longer so much about the final solution as it is an exercise in determining where my "holes" must be in my own understanding of integration. The integral is set up as the summation of 3 integrals actually, so I am going to ask 3 questions, one for each integral.
PART 1 - INTEGRAL 1:
I am trying to complete the following integral:
$∫_0^lcfrac{πR^2}{180} arccosleft(cfrac{R-h+y}{R}right) dx$
where:
$y=sx-sl$
and:
$R, h, s, l - CONSTANT$
The solution I got to was:
$cfrac{πR^3}{180s} left[left(g+cfrac{sx}{R}right)arccos left(g+cfrac{sx}{R}right)-sqrt(1-left(g+cfrac{sx}{R}right)^2) right]|^l_0$
where:
$d=cfrac{R-h}{R}$
$g=d-cfrac{sl}{R}$
Would I be able to get help in determining where I went wrong with this first integral? If you would like to see all of my work, I have saved it in: https://1drv.ms/f/s!AvU6fPuuMW9OpmCbCNz8j_jcFO9j
I will post PART 2 and 3, but am out of time right now!
calculus
asked Jan 27 at 23:06
Royal RougeRoyal Rouge
13
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add a comment |
$begingroup$
I am trying to find a general formula for the volume in a partially filled oblique cylinder (specifically using "dx slices" as the area of the segments). I have set up an integral, which I WAS ABLE to solve numerically and produce correct numerical results, as checked against: https://planetcalc.com/1442/ (their equation set up differently than mine).
So that gives me the ??assurance?? that my integral is a legitimate one. But when I actually complete the integration, it does not produce the correct results. Seeing as I now have a satisfactory solution due to my numerical solving, this exercise is no longer so much about the final solution as it is an exercise in determining where my "holes" must be in my own understanding of integration. The integral is set up as the summation of 3 integrals actually, so I am going to ask 3 questions, one for each integral.
PART 1 - INTEGRAL 1:
I am trying to complete the following integral:
$∫_0^lcfrac{πR^2}{180} arccosleft(cfrac{R-h+y}{R}right) dx$
where:
$y=sx-sl$
and:
$R, h, s, l - CONSTANT$
The solution I got to was:
$cfrac{πR^3}{180s} left[left(g+cfrac{sx}{R}right)arccos left(g+cfrac{sx}{R}right)-sqrt(1-left(g+cfrac{sx}{R}right)^2) right]|^l_0$
where:
$d=cfrac{R-h}{R}$
$g=d-cfrac{sl}{R}$
Would I be able to get help in determining where I went wrong with this first integral? If you would like to see all of my work, I have saved it in: https://1drv.ms/f/s!AvU6fPuuMW9OpmCbCNz8j_jcFO9j
I will post PART 2 and 3, but am out of time right now!
calculus
asked Jan 27 at 23:06
Royal RougeRoyal Rouge
13
$endgroup$
add a comment |
$begingroup$
I am trying to find a general formula for the volume in a partially filled oblique cylinder (specifically using "dx slices" as the area of the segments). I have set up an integral, which I WAS ABLE to solve numerically and produce correct numerical results, as checked against: https://planetcalc.com/1442/ (their equation set up differently than mine).
So that gives me the ??assurance?? that my integral is a legitimate one. But when I actually complete the integration, it does not produce the correct results. Seeing as I now have a satisfactory solution due to my numerical solving, this exercise is no longer so much about the final solution as it is an exercise in determining where my "holes" must be in my own understanding of integration. The integral is set up as the summation of 3 integrals actually, so I am going to ask 3 questions, one for each integral.
PART 1 - INTEGRAL 1:
I am trying to complete the following integral:
$∫_0^lcfrac{πR^2}{180} arccosleft(cfrac{R-h+y}{R}right) dx$
where:
$y=sx-sl$
and:
$R, h, s, l - CONSTANT$
The solution I got to was:
$cfrac{πR^3}{180s} left[left(g+cfrac{sx}{R}right)arccos left(g+cfrac{sx}{R}right)-sqrt(1-left(g+cfrac{sx}{R}right)^2) right]|^l_0$
where:
$d=cfrac{R-h}{R}$
$g=d-cfrac{sl}{R}$
Would I be able to get help in determining where I went wrong with this first integral? If you would like to see all of my work, I have saved it in: https://1drv.ms/f/s!AvU6fPuuMW9OpmCbCNz8j_jcFO9j
I will post PART 2 and 3, but am out of time right now!
calculus
asked Jan 27 at 23:06
Royal RougeRoyal Rouge
13
$endgroup$
I am trying to find a general formula for the volume in a partially filled oblique cylinder (specifically using "dx slices" as the area of the segments). I have set up an integral, which I WAS ABLE to solve numerically and produce correct numerical results, as checked against: https://planetcalc.com/1442/ (their equation set up differently than mine).
So that gives me the ??assurance?? that my integral is a legitimate one. But when I actually complete the integration, it does not produce the correct results. Seeing as I now have a satisfactory solution due to my numerical solving, this exercise is no longer so much about the final solution as it is an exercise in determining where my "holes" must be in my own understanding of integration. The integral is set up as the summation of 3 integrals actually, so I am going to ask 3 questions, one for each integral.
PART 1 - INTEGRAL 1:
I am trying to complete the following integral:
$∫_0^lcfrac{πR^2}{180} arccosleft(cfrac{R-h+y}{R}right) dx$
where:
$y=sx-sl$
and:
$R, h, s, l - CONSTANT$
The solution I got to was:
$cfrac{πR^3}{180s} left[left(g+cfrac{sx}{R}right)arccos left(g+cfrac{sx}{R}right)-sqrt(1-left(g+cfrac{sx}{R}right)^2) right]|^l_0$
where:
$d=cfrac{R-h}{R}$
$g=d-cfrac{sl}{R}$
Would I be able to get help in determining where I went wrong with this first integral? If you would like to see all of my work, I have saved it in: https://1drv.ms/f/s!AvU6fPuuMW9OpmCbCNz8j_jcFO9j
I will post PART 2 and 3, but am out of time right now!
calculus
calculus
asked Jan 27 at 23:06
Royal RougeRoyal Rouge
13
asked Jan 27 at 23:06
Royal RougeRoyal Rouge
13
asked Jan 27 at 23:06
Royal RougeRoyal Rouge
13
asked Jan 27 at 23:06
Royal RougeRoyal Rouge
13
asked Jan 27 at 23:06
Royal RougeRoyal Rouge
13
13
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