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Accepted

### The length of the curve

Let $t:=\operatorname{arsinh}(r/\sqrt2)$ ($0\le t\le\operatorname{arsinh}\sqrt2$). Then, $$r=\sqrt2\sinh t,\quad r'_t=\sqrt2\cosh t,$$$$\theta=\sinh t\cosh t+t+\ln\sqrt2,\quad\theta'_t=2\cosh^2t,$$ ...
• 37.4k
Accepted

### Clarifications on the solution of a double integral: $\iint_X\frac{x^2y}{x^2+y^2}dxdy$

You can transform the conditions to polar coordinates as well. From the first condition, you get $r^2\cos^2\vartheta+r^2\sin^2\vartheta\geq1$, which reduces to $r\geq1$. From the second condition, you ...
• 2,699

### Solving Laplace's equation on semi-annular domain

The solution form you have written is for a disk not an annulus or half-annulus Separation of variables can be used in your half-annulus too, and the form you will get is A + B\ln r + \sum_{n=1}^{\...
• 45.3k
1 vote
Accepted

### Average distance from a point on a circle to the y-axis.

On the circle $x^2+y^2=9$ we have identically $r\equiv 3$, so in the integral we should consider the length of the circle (which is $2\pi r=6\pi$) and integrate only on $\theta$. Thus, the integral is ...
• 7,832

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