I want to integrate a function over a region R, which is determined by the following conditions. $$x^2+y^2\leq1$$ and $$x+y\geq1$$

My region is determined by these two inequalities. I know that the first expression produces a unit circle, which contains all points within the unit circle, including the boundaries.The second expression is a line passing through $y=1$ and with a slope of $-1$.

The region is given to be between the unit circle and the line .I do know what the region looks like , but I do not know how to express it in terms of integration limits, so I can integrate it and find the area, or just do a general integration over this region. I tried to use polar coordinates, but still couldn't figure it out.

Can you guys help me determine the integration limits for this region ?

  • $\begingroup$ What function will you be integrating? $\endgroup$ – John Lou Nov 6 '17 at 21:11
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    $\begingroup$ Do you prefer to use polar coordinates? This can be done with rectangular or polar... $\endgroup$ – G Tony Jacobs Nov 6 '17 at 21:11
  • $\begingroup$ @JohnLou the integrand of my integral is given by dxdy / (x^2+y^2)^2 $\endgroup$ – Viktor Raspberry Nov 6 '17 at 21:12
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    $\begingroup$ Since the integrand looks like $1/r^4,$ I'd probably try polar coordinates first. Do you know how to write the equation of the line $y=1-x$ in polar coordinates? $\endgroup$ – David K Nov 6 '17 at 21:14
  • $\begingroup$ $x^2 + y^2 \le 1$ is the area inside the circle of unit radius centred. $x + y \ge 1$ is the area above the line $x + y = 1$. Find the intersection between circle and line. $\endgroup$ – A---B Nov 6 '17 at 21:14

Carrying out this integral in polar coordinates, it is first clear that we need to integrate from $\theta=0$ to $\theta=\frac{\pi}{2}$. The lower limit for $r$ is given by the line, and the upper limit is given by the circle $r=1$. Thus, we need to way to express the line $x+y=1$ in polar coordinates. Using the usual substitution, we can write:




Thus, it looks like we can integrate:

$$\int_0^{\frac{\pi}{2}}\int_{\frac{1}{\cos\theta+\sin\theta}}^1 f(r\cos\theta,r\sin\theta)r\,dr\,d\theta$$


enter image description here

$\int_0^1 \int_{y}^{\sqrt {1-y^2}} f(x,y)\ dx\ dy$ would be one way.

Of course it is symmetric,

$\int_0^1 \int_{x}^{\sqrt {1-x^2}} f(x,y)\ dy\ dx$ would also work.

or you can convert to polar

$x = r\cos\theta\\ y = r\sin \theta\\ dy\ dx = r\ dr\ d\theta$

$x+y \ge1\\ r\cos\theta + r\sin\theta \ge 1\\ r\sqrt 2(\cos\theta - \frac {\pi}{4}) = 1\\ r = \frac {\sqrt 2}{2} \sec (\theta-\frac {\pi}{4})$

$\int_0^{\frac {\pi}{2}} \int_{\sec(\theta - \frac {\pi}{4})}^1 rf(r,\theta)\ dr\ d\theta$


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