🎓 Gabarito Comentado (AVA):

𝐼

∮ 𝐶 𝐹 · 𝑑 𝑟

∬ 𝐷 ∂ 𝐹 2 ∂ 𝑥

∂ 𝐹 1 ∂ 𝑦 𝑑 𝐴 ∂ 𝐹 2 ∂ 𝑥

∂ ∂ 𝑥

𝑦 𝑥 2 + 𝑦 2

( 0 ) ( 𝑥 2 + 𝑦 2 )

2 𝑥 (

𝑦 ) ( 𝑥 2 + 𝑦 2 ) 2

2 𝑥 𝑦 ( 𝑥 2 + 𝑦 2 ) 2 ∂ 𝐹 1 ∂ 𝑥

∂ ∂ 𝑥

𝑦 𝑥 2 + 𝑦 2

( 0 ) ( 𝑥 2 + 𝑦 2 )

2 𝑥 ( 𝑦 ) ( 𝑥 2 + 𝑦 2 ) 2

2 𝑥 𝑦 ( 𝑥 2 + 𝑦 2 ) 2 ∂ 𝐹 2 ∂ 𝑥

∂ 𝐹 1 ∂ 𝑦

4 𝑥 𝑦 ( 𝑥 2 + 𝑦 2 ) 2 𝐼

∮ 𝐶 𝐹 ∙ 𝑑 𝑟

∬ 𝐷 4 𝑥 𝑦 ( 𝑥 2 + 𝑦 2 ) 2 𝑑 𝐴

Entendendo a D:

Fonte: Elaborada pelo(a) autor(a).

Direções polares:

{ 𝑥

𝑟 𝑐 𝑜 𝑠 𝜃 𝑦

𝑟 𝑠 𝑒 𝑛 𝜃 𝐽

𝑟 , onde  𝑟 ∈ [ 1 , 2 ] e π π 𝜃 ∈ 0 , π 2

π π π π π π π π π π ∬ 𝐷 4 𝑥 𝑦 ( 𝑥 2 + 𝑦 2 ) 2 𝑑 𝐴

∫ 0 π / 2 ∫ 1 2 4 ( 𝑟 𝑐 𝑜 𝑠 𝜃 ) ( 𝑟 𝑠 𝑒 𝑛 𝜃 ( 𝑟 𝑐 𝑜 𝑠 𝜃 ) 2 + ( 𝑟 𝑠 𝑒 𝑛 𝜃 ) 2 ) 2 ) + 𝑑 𝑟 𝑑 𝜃 ∫ 0 π / 2 ∫ 1 2 4 𝑟 2 𝑐 𝑜 𝑠 𝜃 𝑠 𝑒 𝑛 𝜃 · 𝑟 ( 𝑟 2 𝑐 𝑜 𝑠 2 𝜃 + 𝑟 2 𝑠 𝑒 𝑛 2 𝜃 ) 2 𝑑 𝑟 𝑑 𝜃

∫ 0 π / 2 ∫ 1 2 4 𝑟 3 𝑐 𝑜 𝑠 𝜃 𝑠 𝑒 𝑛 𝜃 ( 𝑟 2 ( 𝑐 𝑜 𝑠 2 𝜃 + 𝑠 𝑒 𝑛 2 𝜃 ) ) 2 𝑑 𝑟 𝑑 𝜃

= ∫ 0 π / 2 ∫ 1 2 4 𝑐 𝑜 𝑠 𝜃 𝑠 𝑒 𝑛 𝜃 𝑟 𝑑 𝑟 𝑑 𝜃

∫ 0 π / 2 ∫ 1 1 2 𝑠 𝑒 𝑛 2 𝜃 𝑟 𝑑 𝑟 𝑑 𝜃

Integrando: π π π π π π π π π π ∫ 0 π / 2 2 𝑠 𝑒 𝑛 2 𝜃 [ 𝑙 𝑛 𝑟 ]   𝑟

1 2 𝑑 𝜃

∫ 0 π / 2 2 𝑠 𝑒 𝑛 2 𝜃 [ ( 𝑙 𝑛 2 )

( 𝑙 𝑛 1 ) ] 𝑑 𝜃

2 𝑙 𝑛 2 ∫ 0 π / 2 𝑠 𝑒 𝑛 2 𝜃 𝑑 𝜃 2 𝑙 𝑛 2

2 𝑐 𝑜 𝑠 2 𝜃 2 𝜃

0 π / 2

2 𝑙 𝑛 2

𝑐 𝑜 𝑠 2 π 2 2

𝑐 𝑜 𝑠 2 · 0 2 2

2 𝑙 𝑛 2 1 2 + 1 2

2 𝑙 𝑛 2 .