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Posted: 4/21/2021 11:24:07 AM EDT
Can anyone tell me if I’m setting this integral up correctly? I am trying to determine the area of the shaded part of this circle using integration. I put my integral into an online solver but I’m not 100% if I have set up the integral right. Any math genius’s out there that can comment?
I’m trying to avoid doing this one with respect to y because the solution has some pretty complex trig functions and I would like to solve for h and that’s tough to do when it’s buried under inverse trig functions. Any comments or help is appreciated. 
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Posted: 4/21/2021 2:03:35 PM EDT
[#1]
Set up the equation for the infinitesimal area dx mulitiplied by its non constant height; I'll call that k(x). dA= k(x)dx
First work out the problem with h=0, then extend that solution to h>0. The other problem in your solution is switching from a solution for y, to suddenly substituting x. Integration is nothing more than summing. A bunch of infinitesimal rectangles in this case. h(x) is not a continuous equation due to the curvature at the ends of the constant height part. The limits of integration of the upper segment are not -r to +r. That is not a more simple method. Integrate on dy from h to r. Calculate the width of dy from the equation for a circle at y = h. |
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Posted: 4/21/2021 8:35:06 PM EDT
[#2]
Quote HistoryQuoted: Set up the equation for the infinitesimal area dx mulitiplied by its non constant height; I'll call that k(x). dA= k(x)dx First work out the problem with h=0, then extend that solution to h>0. The other problem in your solution is switching from a solution for y, to suddenly substituting x. Integration is nothing more than summing. A bunch of infinitesimal rectangles in this case. h(x) is not a continuous equation due to the curvature at the ends of the constant height part. The limits of integration of the upper segment are not -r to +r. That is not a more simple method. Integrate on dy from h to r. Calculate the width of dy from the equation for a circle at y = h. Thank you for your input. You are correct solving using a horizontal rectangle with thickness dy is an easier process. the result it churns up is just a little nasty, as I want to isolate h and its stuck under a bunch of inverse trig functions. I ended up putting that equation in excel and using the "goal seek" function to solve for h. it works Any idea how to isolate D in this equation, can a solver like MatLab do this?  |
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Posted: 4/22/2021 2:31:52 AM EDT
[#3]
Look at your limits of integration. Does the line y=h intersect the semicircle when x = r or x = -r?
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