- #1

VinnyCee

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Another problem that I cannot figure out. Convert the follorwing into polar coordinates:

[tex]\int_{-1}^{1} \int_{-\sqrt{1 - y^2}}^{\sqrt{1 - y^2}} ln\left(x^2 + y^2 + 1\right) dx\;dy[/tex]

I did this so far:

[tex]ln\left(x^2 + y^2 + 1\right) = ln\left(r^2 + 1\right)[/tex]

[tex]\sqrt{1 - y^2} = \sqrt{1 - r^2 \sin^2 \theta}[/tex]

Now what do I do?

Is this possibly right?

[tex]\int_{0}^{2\pi} \int_{-\sqrt{1 - r^2 \sin^2 \theta}}^{\sqrt{1 - r^2 \sin^2 \theta}} ln\left(r^2 + 1\right) dr\;d\theta[/tex]

[tex]\int_{-1}^{1} \int_{-\sqrt{1 - y^2}}^{\sqrt{1 - y^2}} ln\left(x^2 + y^2 + 1\right) dx\;dy[/tex]

I did this so far:

[tex]ln\left(x^2 + y^2 + 1\right) = ln\left(r^2 + 1\right)[/tex]

[tex]\sqrt{1 - y^2} = \sqrt{1 - r^2 \sin^2 \theta}[/tex]

Now what do I do?

Is this possibly right?

[tex]\int_{0}^{2\pi} \int_{-\sqrt{1 - r^2 \sin^2 \theta}}^{\sqrt{1 - r^2 \sin^2 \theta}} ln\left(r^2 + 1\right) dr\;d\theta[/tex]

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