For the region R below, write \int \int_Z f dA

Question

babeeb0oL 2021-09-11 Answered

For the region R below, write $\int \int_{Z} f d A$ as an integral in polar coordinates.

Use t for $θ$ in your expressions.

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Expert Answer

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 and x^{2} + y^{2} = 16

Therefore, region R can be defined as

R = {{(r, θ)}^{3} \leq r \leq 4, 0 \leq θ \leq π}

it is known that,

\int \int_{R} f d A = \int_{θ_{1}}^{θ_{2}} f (r \cos θ, r \sin θ) r d r d θ

Therefore, in polar co-ordinate,

\int \int_{R} f d A = \int_{0}^{π} \int_{3}^{4} f (r \cos θ, r \sin θ) r d r d θ

(i)
Thus, comparing (i) with

\int \int_{R} f d A = \int_{0}^{π} \int_{3}^{4} f (r \cos θ, r \sin θ) r d r d θ

it is inferred that
a = 0

b = π

c = 3
d = 4
With

d A = r d r d θ

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