Given-that-I-R-x-2-y-2-5-2-dxdy-where-R-is-the-region-x-2-y-2-a-2-use-a-suitable-transformation-to-evaluate-I-

Question Number 204500 by DeArtist last updated on 19/Feb/24

Given that I = \int \int_{R} {(x^{2} + y^{2})}^{\frac{5}{2}} d x d y where R

is the region x^{2} + y^{2} ⩽ a^{2}

use a suitable transformation to evaluate I

Answered by witcher3 last updated on 19/Feb/24

x^{2} + y^{2} ⩽ a^{2} \Rightarrow (x = rcos (t); y = rsin (t))

r \in [0, ∣ a ∣] t \in [0, 2 π]

I = \int_{0}^{2 π} \int_{0}^{∣ a ∣} r^{5} . rdrdt

= \int_{0}^{2 π} \int_{0}^{∣ a ∣} r^{6} dr . dt = \int_{0}^{2 π} dt . \int_{0}^{∣ a ∣} r^{6} dr

= \frac{2 π}{7} . ∣ a ∣^{7}

Answered by mr W last updated on 19/Feb/24

x = r \cos θ

y = r \sin θ

d x d y = r d θ d r

I = \int_{0}^{a} \int_{0}^{2 π} r^{6} d θ d r = \frac{2 π a^{7}}{7}

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