2 can be evaluated as 2 in polar coordinates 2 2

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Unformatted text preview: esian coordinates and as 0 2 π 6 2 0 π2 1 + cos 6θ θ= 2 4 ∞ 2 E S cos 3θ θ = 3 2 2− ∞ 0 and use it to compute −∞ − . 2 ∞ ∞ can be evaluated as −∞ −∞ −2 θ in polar coordinates). :) 2 − 2− 2 in In polar coordinates, we have − 2π 2− 2 ∞ − − = R2 0 ∞ 2 θ = 2π − 2 =π 2 0 0 On the other hand, ∞ π= and hence ∞ ∞ − −∞ ∞ −∞ − 2− 2 = −∞ √ 2 = ∞ − 2 −∞ · −∞ π. ∞ − − 2 = 2 2 −∞ :) 6 E ing solids: 2 (a) (b) ( (c) ( S + 2 2 + + Applying cylindrical or spherical coordinates, find the volume of the follow2 + 2 2 + + ≤2 , )≤ 2 2 22 2 2 ( ) ≤3 22 2 + ≤ 2 2 + − 2 > 0. , where ). . (a) Both spherical and cylindrical coordinates can be applied. In cylindrical coordinates, we have 2 + 2 ≤ 2 and 2 ≤ 2 , that is, 2 + ( − )2 ≤ 2 and ( − )( + ) ≥ 0. Since ≥ 0 (as this follows from 2 ≥ 2 + 2 + 2 ≥ 0) and ≥ 0, it follows that ≥ . Moreover, the condition 2 + ( − )2 ≤ 2 implies that ( − √ )2 ≤ 2 − 2 which√...
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