Integrating $\int_1^2 \int_0^ \sqrt{2x-x^2} \frac{1}{((x^2+y^2)^2} dydx $ in polar coordinates

5,707

Solution 1

I think you will find it easiest if you write the integral in the form $$\int_?^?\int_?^? \frac{1}{r^3}\,dr\,d\theta\ ,$$ that is, with $\theta$ on the "outside" integral. Looking at your diagram, the minimum $\theta$ value in the region of integration is $\theta=0$, which occurs along the $x$ axis. The maximum $\theta$ value occurs at the point $(1,1)$, giving $\theta=\pi/4$. So we have $$\int_0^{\pi/4}\int_?^? \frac{1}{r^3}\,dr\,d\theta\ .$$ For any fixed $\theta$ value, the values of $r$ in the region go from the vertical line $x=1$ to the semicircle. At $x=1$ we have $r=\sec\theta$ as you have noted. The easiest way to get the maximum $r$ value is to draw a line from the origin at angle $\theta$ until it hits the semicircle, and a line from there to $(2,0)$. Then you can see in the diagram a right-angled triangle (because the angle in a semicircle is a right angle) with hypotenuse $2$, and so we get $r_{\rm max}=2\cos\theta$. So the integral is $$\int_0^{\pi/4}\int_{\sec\theta}^{2\cos\theta} \frac{1}{r^3}\,dr\,d\theta\ .$$

For sketching the graph, $$y=\sqrt{2x-x^2}\quad\Rightarrow\quad x^2-2x+y^2=0 \quad\Rightarrow\quad (x-1)^2+y^2=1\ ,$$ and not forgetting that $y\ge0$.

Solution 2

$$ \sqrt{2\,x-x^2}=\sqrt{1-(x-1)^2}. $$ The graph is a semicircle of radius $1$ centered at $(1,0)$.

Share:
5,707

Related videos on Youtube

user125342
Author by

user125342

Updated on February 18, 2022

Comments

  • user125342
    user125342 over 1 year

    I'm having a problem converting $\int\limits_1^2 \int\limits_0^ \sqrt{2x-x^2} \frac{1}{(x^2+y^2)^2} dy dx $ to polar coordinates.

    I drew the graph using my calculator, which looked like half a circle on the x axis. I know that $\frac{1}{(x^2+y^2)^2} dydx$ turns to $ \frac{r}{(r^2)^2}drd\theta$, which would be $ \frac{1}{r^3}$

    The region of integration in $\theta $ is I guess $0 \leq \theta \leq \frac{1}{4} \pi $, but I'm stuck for r. The lower boundary is $x=1=rcos\theta \rightarrow r=1/cos\theta=sec\theta$, but what is the upper boundary?? Is it $\sqrt{2rcos\theta-r^2cos^2\theta}$? I tried this but I couldn't integrate my equation...

    Could someone please help me out?

    Also, how can I draw the graph of $y= \sqrt{2x-x^2} $? I could only know how it looked like by using my calculator...

  • user125342
    user125342 over 9 years
    ok so i understand how you got a right triangle with hypotenuse 2, but how did that become $2cos \theta $..??
  • David
    David over 9 years
    Because the included angle is $\theta$ and the adjacent side is $r_{\rm max}$. Extract the triangle from the diagram and draw it separately if that helps.
  • user125342
    user125342 over 9 years
    ohhhh ok thank you so much!! I didn't know i can find rmax like that!!
  • David
    David over 9 years
    You could also get it from writing $y^2=2x-x^2$ as$$r^2\sin^2\theta=2r\cos\theta-r^2\cos^2\theta\ ,$$but I think the geometric way is neater.