Using a Surface Integral for Mathematical Analysis of the Area of an Island

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The discussion focuses on using a surface integral to analyze the area of an island shaped like an upside-down paraboloid. There is confusion regarding the integration limits for the variable r, with advice given to reverse them to ensure a positive result. A link to a relevant Wikipedia page on spherical caps is provided for further clarification. Participants agree on the necessity of adjusting the integration limits. The conversation emphasizes the importance of correctly setting up the integral for accurate mathematical analysis.
daphnelee-mh
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Relevant Equations
A=sqrt[1+(dz/dx)^2+(dz/dy)^2]dA
1593786915216.png

I am not clearly understand what the question requests for, is it okay to continue doing like this ? Kindly advise, thanks
 
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The shape of the island is an upside-down paraboloid. The integral looks like everything is OK except the ##r## limits should be reversed. You want to integrate from the smallest to largest values of ##r## to get a positive answer.
 
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LCKurtz said:
The shape of the island is an upside-down paraboloid. The integral looks like everything is OK except the ##r## limits should be reversed. You want to integrate from the smallest to largest values of ##r## to get a positive answer.
I believe you're right. I looked at it wrong.
 

Thread 'Does this series converge uniformly?'

First, I tried to show that ##f_n## converges uniformly on ##[0,2\pi]##, which is true since ##f_n \rightarrow 0## for ##n \rightarrow \infty## and ##\sigma_n=\mathrm{sup}\left| \frac{\sin\left(\frac{n^2}{n+\frac 15}x\right)}{n^{x^2-3x+3}} \right| \leq \frac{1}{|n^{x^2-3x+3}|} \leq \frac{1}{n^{\frac 34}}\rightarrow 0##. I can't use neither Leibnitz's test nor Abel's test. For Dirichlet's test I would need to show, that ##\sin\left(\frac{n^2}{n+\frac 15}x \right)## has partialy bounded sums...
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