Sketch the region of integration and evaluate the following integral. ModifyingBelow Integral from nothing to nothing Integral from nothing

Question

Sketch the region of integration and evaluate the following integral. ModifyingBelow Integral from nothing to nothing Integral from nothing to nothing With Upper R StartFraction 1 Over 3 plus StartRoot x squared plus y squared EndRoot EndFraction dA ​, RequalsStartSet (r comma theta ): 0 less than or equals r less than or equals 2 comma StartFraction pi Over 2 EndFraction less than or equals theta less than or equals StartFraction 3 pi Over 2 EndFraction EndSet

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Liliana 1 week 2021-10-08T05:42:39+00:00 1 Answer 0

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    2021-10-08T05:44:03+00:00

    Answer:

    \frac{10\pi}{3}

    Step-by-step explanation:

    According to the information of the problem we have to compute the following integral.

    {\displaystyle \int\limits \int} \frac{1}{3} + \sqrt{x^2 + y^2}  \, dA

    Where the region of integration is

    R = \Big\{ (r,\theta) :  0 \leq r \leq 2 , \,\,\,\, \frac{\pi}{2} \leq \theta \leq \frac{3\pi}{2} \Big\}

    If you plot, that is just a circle between \pi/2  and  3\pi/2, which is just half of the circle on the negative part of the plane.

    When you switch coordinates

    {\displaystyle \int\limits \int} \frac{1}{3} + \sqrt{x^2 + y^2}  \, dA  = {\displaystyle \int\limits_{0}^{2} \int\limits_{\pi/2}^{3\pi/2}} \bigg(\frac{1}{3} + r \bigg)r \, d\theta\, dr  = \frac{10\pi}{3}

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