For the region R below, write

Use t for

babeeb0oL
2021-09-11
Answered

For the region R below, write

Use t for

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estenutC

Answered 2021-09-12
Author has **81** answers

From the given figure, it is seen that the region R is bounded by two circles, ${x}^{2}+{y}^{2}=9{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}{x}^{2}+{y}^{2}=16$

Therefore, region R can be defined as

$R=\{{(r,\theta )}^{3}\le r\le 4,0\le \theta \le \pi \}$

it is known that,

$\int {\int}_{R}fdA={\int}_{{\theta}_{1}}^{{\theta}_{2}}f(r\mathrm{cos}\theta ,r\mathrm{sin}\theta )rdrd\theta$

Therefore, in polar co-ordinate,

$\int {\int}_{R}fdA={\int}_{0}^{\pi}{\int}_{3}^{4}f(r\mathrm{cos}\theta ,r\mathrm{sin}\theta )rdrd\theta$ (i)

Thus, comparing (i) with$\int {\int}_{R}fdA={\int}_{0}^{\pi}{\int}_{3}^{4}f(r\mathrm{cos}\theta ,r\mathrm{sin}\theta )rdrd\theta$ it is inferred that

a = 0

$b=\pi$

c = 3

d = 4

With$dA=rdrd\theta$

Therefore, region R can be defined as

it is known that,

Therefore, in polar co-ordinate,

Thus, comparing (i) with

a = 0

c = 3

d = 4

With

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