𝑬𝒄𝒖𝒂𝒄𝒊𝒐𝒏𝒆𝒔 𝒑𝒂𝒓𝒂𝒎é𝒕𝒓𝒊𝒄𝒂𝒔 ∗ 𝑫𝒆𝒓𝒊𝒗𝒂𝒅𝒂 𝒆𝒏 𝒇𝒐𝒓𝒎𝒂 𝒑𝒂𝒓𝒂𝒎é𝒕𝒓𝒊𝒄𝒂 𝑇𝑜𝑑𝑎 𝑐𝑢𝑟𝑣𝑎 𝑑𝑎𝑑𝑎 𝑝𝑜𝑟 𝑥 = 𝑓(𝑡), 𝑒 𝑦 = 𝑔(𝑡), 𝑙𝑎 𝑝𝑒𝑛𝑑𝑖𝑒𝑛𝑡𝑒 𝑒𝑛 𝑒𝑛 (𝑥, 𝑦) 𝑒𝑠: 𝑑𝑦 𝑑𝑦 𝑑𝑡 𝑑𝑥 = , 𝑐𝑜𝑛 ≠0 𝑑𝑥 𝑑𝑥 𝑑𝑡 𝑑𝑡 𝑒𝑛 𝑒𝑙 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙𝑜 [𝑎, 𝑏], 𝑠𝑢 𝑙𝑜𝑛𝑔𝑖𝑡𝑢𝑑 𝑑𝑒 ∗ 𝑳𝒐𝒏𝒈𝒊𝒕𝒖𝒅 𝒅𝒆 𝒂𝒓𝒄𝒐 𝒆𝒏 𝒇𝒐𝒓𝒎𝒂 𝒑𝒂𝒓𝒂𝒎é𝒕𝒓𝒊𝒄𝒂 𝑇𝑜𝑑𝑎 𝑐𝑢𝑟𝑣𝑎 𝑑𝑎𝑑𝑎 𝑝𝑜𝑟 𝑥 = 𝑓(𝑡), 𝑒 𝑦 = 𝑔(𝑡), 𝑒𝑛 𝑒𝑙 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙𝑜 [𝑎, 𝑏], 𝑠𝑢 𝑙𝑜𝑛𝑔𝑖𝑡𝑢𝑑 𝑑𝑒 𝑎𝑟𝑐𝑜 𝑣𝑖𝑒𝑛𝑒 𝑑𝑎𝑑𝑎 𝑝𝑜𝑟: 𝑑𝑥 2 𝑑𝑦 2 √ 𝐿 = ∫ ( ) + ( ) 𝑑𝑡 𝑑𝑡 𝑑𝑡 𝑎 𝑏
∗ 𝑨𝒓𝒆𝒂 𝒅𝒆 𝒖𝒏𝒂 𝒔𝒖𝒑𝒆𝒓𝒇𝒊𝒄𝒊𝒆 𝒅𝒆 𝒓𝒆𝒗𝒐𝒍𝒖𝒄𝒊ó𝒏 𝑇𝑜𝑑𝑎 𝑐𝑢𝑟𝑣𝑎 𝑑𝑎𝑑𝑎 𝑝𝑜𝑟 𝑥 = 𝑓(𝑡), 𝑒 𝑦 = 𝑔(𝑡) 𝑒𝑛 𝑒𝑙 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙𝑜 [𝑎, 𝑏], 𝑒𝑙 á𝑟𝑒𝑎 𝑺 𝑑𝑒 𝑙𝑎 𝑠𝑢𝑝𝑒𝑟𝑓𝑖𝑐𝑖𝑒 𝑑𝑒 𝑟𝑒𝑣𝑜𝑙𝑢𝑐𝑖ó𝑛 𝑜𝑏𝑡𝑒𝑛𝑖𝑑𝑎 𝑣𝑖𝑒𝑛𝑒 𝑑𝑎𝑑𝑎 𝑝𝑜𝑟 𝑝𝑜𝑠: 𝑑𝑥 2 𝑑𝑦 2 √ 𝑆 = 2π ∫ 𝑔(𝑡) ( ) + ( ) 𝑑𝑡 𝑑𝑡 𝑑𝑡 𝑎 𝑏
𝑒𝑗𝑒 𝑑𝑒 𝑟𝑒𝑣𝑜𝑙𝑢𝑐𝑖ó𝑛 𝑥; 𝑔(𝑡) ≥ 0
𝑑𝑥 2 𝑑𝑦 2 √ 𝑆 = 2π ∫ 𝑓(𝑡) ( ) + ( ) 𝑑𝑡 𝑒𝑗𝑒 𝑑𝑒 𝑟𝑒𝑣𝑜𝑙𝑢𝑐𝑖ó𝑛 𝑦; 𝑓(𝑡) ≥ 0 𝑑𝑡 𝑑𝑡 𝑎 𝑪𝒐𝒐𝒓𝒅𝒆𝒏𝒂𝒅𝒂𝒔 𝒑𝒐𝒍𝒂𝒓𝒆𝒔 𝐿𝑎𝑠 𝑐𝑜𝑜𝑟𝑑𝑒𝑛𝑎𝑑𝑎𝑠 𝑝𝑜𝑙𝑎𝑟𝑒𝑠 (𝑟, ѳ)𝑑𝑒 𝑢𝑛 𝑝𝑢𝑛𝑡𝑜 𝑒𝑠𝑡á𝑛 𝑟𝑒𝑙𝑎𝑐𝑖𝑜𝑛𝑎𝑑𝑎𝑠 𝑐𝑜𝑛 𝑠𝑢𝑠 𝑐𝑜𝑜𝑟𝑑𝑒𝑛𝑎𝑑𝑎𝑠 𝑟𝑒𝑐𝑡𝑎𝑛𝑔𝑢𝑙𝑎𝑟𝑒𝑠 (𝑥, 𝑦)𝑝𝑜𝑟: 𝑥 = 𝑟 ∗ cos(ѳ) 𝑦 = 𝑟 ∗ 𝑠𝑒𝑛(ѳ) 𝑦 𝑡𝑔 (ѳ) = 𝑟2 = 𝑥2 + 𝑦2 𝑥 𝑫𝒆𝒓𝒊𝒗𝒂𝒅𝒂 𝒆𝒏 𝒇𝒐𝒓𝒎𝒂 𝒑𝒐𝒍𝒂𝒓 𝑆í 𝑓 𝑒𝑠 𝑢𝑛𝑎 𝑓𝑢𝑛𝑐𝑖ó𝑛 𝑑𝑒𝑟𝑖𝑣𝑎𝑏𝑙𝑒 𝑑𝑒 ѳ, 𝑙𝑎 𝑝𝑒𝑛𝑑𝑖𝑒𝑛𝑡𝑒 𝑑𝑒 𝑙𝑎 𝑟𝑒𝑐𝑡𝑎 𝑡𝑎𝑛𝑔𝑒𝑛𝑡𝑒 𝑎 𝑙𝑎 𝑔𝑟á𝑓𝑖𝑐𝑎 𝑑𝑒 𝑟 = 𝑓(ѳ) 𝑒𝑛 𝑒𝑙 𝑝𝑢𝑛𝑡𝑜 (𝑟, ѳ)𝑒𝑠: 𝑑𝑦 𝑑𝑦 𝑑ѳ 𝑑𝑥 = , 𝑐𝑜𝑛 ≠ 0 𝑒𝑛 (𝑟, ѳ) 𝑑𝑥 𝑑𝑥 𝑑ѳ 𝑑ѳ 𝑨𝒓𝒆𝒂 𝒆𝒏 𝒄𝒐𝒐𝒓𝒅𝒆𝒏𝒂𝒅𝒂𝒔 𝒑𝒐𝒍𝒂𝒓𝒆𝒔 𝑆í 𝑓 𝑒𝑠 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑎 𝑦 𝑛𝑜 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑎 𝑒𝑛 𝑒𝑙 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙𝑜 [𝛼, 𝛽], 𝑒𝑙 á𝑟𝑒𝑎 𝑑𝑒 𝑙𝑎 𝑟𝑒𝑔𝑖ó𝑛 𝑙𝑖𝑚𝑖𝑡𝑎𝑑𝑎 𝑝𝑜𝑟 𝑙𝑎 𝑔𝑟á𝑓𝑖𝑐𝑎 𝑑𝑒 𝑟 = 𝑓(ѳ)𝑦 𝑙𝑎𝑠 𝑟𝑒𝑐𝑡𝑎𝑠 𝑟𝑎𝑑𝑖𝑎𝑙𝑒𝑠 ѳ = 𝛼 𝑦 ѳ = 𝛽 𝑒𝑠𝑡á 𝑑𝑎𝑑𝑎 𝑝𝑜𝑟: 1 𝛽 1 𝛽 2 2 𝐴 = ∫ [𝑓(ѳ)] 𝑑ѳ = ∫ 𝑟 𝑑ѳ 2 𝛼 2 𝛼 ∗ 𝑳𝒐𝒏𝒈𝒊𝒕𝒖𝒅 𝒅𝒆 𝒂𝒓𝒄𝒐 𝒆𝒏 𝒇𝒐𝒓𝒎𝒂 𝒑𝒂𝒓𝒂𝒎é𝒕𝒓𝒊𝒄𝒂 𝑆𝑒𝑎 𝑓 𝑢𝑛𝑎 𝑓𝑢𝑛𝑐𝑖ó𝑛 𝑐𝑢𝑦𝑎 𝑑𝑒𝑟𝑖𝑣𝑎𝑑𝑎 𝑒𝑠 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑎 𝑒𝑛 𝑢𝑛 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙𝑜 𝛼 ≤ 𝜃 ≤ 𝛽. 𝐿𝑎 𝑙𝑜𝑛𝑔𝑖𝑡𝑢𝑑 𝑑𝑒 𝑎𝑟𝑐𝑜 𝑑𝑒 𝑙𝑎 𝑔𝑟á𝑓𝑖𝑐𝑎 𝑑𝑒 𝑟 = 𝑓(𝜃)𝑑𝑒𝑠𝑑𝑒 𝜃 = 𝛼 ℎ𝑎𝑠𝑡𝑎 𝜃 = 𝛽 𝑒𝑠: 𝑏
𝑑𝑟 2 ] 𝑑𝜃 𝑑𝜃 𝛼 𝛼 ∗ 𝑨𝒓𝒆𝒂 𝒅𝒆 𝒖𝒏𝒂 𝒔𝒖𝒑𝒆𝒓𝒇𝒊𝒄𝒊𝒆 𝒅𝒆 𝒓𝒆𝒗𝒐𝒍𝒖𝒄𝒊ó𝒏 𝛽
𝛽
𝑠 = ∫ √[𝑓(𝜃)]2 + [𝑓´(𝜃)]2 𝑑𝜃 = ∫ √𝑟 2 + [
𝑆𝑒𝑎 𝑓 𝑢𝑛𝑎 𝑓𝑢𝑛𝑐𝑖ó𝑛 𝑐𝑢𝑦𝑎 𝑑𝑒𝑟𝑖𝑣𝑎𝑑𝑎 𝑒𝑠 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑎 𝑒𝑛 𝑢𝑛 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙𝑜 𝛼 ≤ 𝜃 ≤ 𝛽, 𝐸𝑙 á𝑟𝑒𝑎 𝑑𝑒 𝑙𝑎 𝑠𝑢𝑝𝑒𝑟𝑓𝑖𝑐𝑖𝑒 𝑔𝑒𝑛𝑒𝑟𝑎𝑑𝑎 𝑝𝑜𝑟 𝑟𝑒𝑣𝑜𝑙𝑢𝑐𝑖ó𝑛 𝑑𝑒 𝑙𝑎 𝑔𝑟á𝑓𝑖𝑐𝑎 𝑑𝑒 𝑟 = 𝑓(𝜃), 𝑑𝑒𝑠𝑑𝑒 𝜃 = 𝛼, ℎ𝑎𝑠𝑡𝑎 𝜃 = 𝛽 𝑒𝑛 𝑡𝑜𝑟𝑛𝑜 𝑑𝑒 𝑙𝑎 𝑟𝑒𝑐𝑡𝑎 𝑖𝑛𝑑𝑖𝑐𝑎𝑑𝑎 𝑒𝑠 𝑙𝑎 𝑠𝑖𝑔𝑢𝑖𝑒𝑛𝑡𝑒: 𝛽
𝑆 = 2𝜋 ∫ 𝑓(𝜃)𝑠𝑒𝑛𝜃√[𝑓(𝜃)]2 + [𝑓´(𝜃)]2 𝑑𝜃 𝑒𝑛 𝑡𝑜𝑟𝑛𝑜 𝑎𝑙 𝑒𝑗𝑒 𝑝𝑜𝑙𝑎𝑟 𝛼 𝛽
𝑆 = 2𝜋 ∫ 𝑓(𝜃)𝑐𝑜𝑠𝜃√[𝑓(𝜃)]2 + [𝑓´(𝜃)]2 𝑑𝜃 𝑒𝑛 𝑡𝑜𝑟𝑛𝑜 𝑎 𝑙𝑎 𝑟𝑒𝑐𝑡𝑎 𝜃 = 𝛼
𝜋 2
𝑇𝑎𝑙𝑙𝑒𝑟 𝐶á𝑙𝑐𝑢𝑙𝑜 𝑀𝑢𝑙𝑡𝑖𝑣𝑎𝑟𝑖𝑎𝑑𝑜 𝐺𝑟𝑎𝑓𝑖𝑐𝑎𝑟 1. 𝑟(ѳ) = 3 𝜋 2. ѳ = 6 3. 𝑟(ѳ) = 3ѳ 4. 𝑟(ѳ) = 0.5 + cos(ѳ) 3 5. 𝑟(ѳ) = + 2 ∗ cos(ѳ) 2 6. 𝑟(ѳ) = 2 ∗ 𝑠𝑒𝑐(ѳ) + 3 5 7. 𝑟(ѳ) = 2 − 2𝑠𝑒𝑛(ѳ) 8. 𝑟(ѳ) = 16 ∗ cos(2ѳ) ѳ 9. 𝑟(ѳ) = 1 + 3 ∗ 𝑠𝑒𝑛 ( ) 2 10. 𝑟(𝜃) = 1 − 3 ∗ 𝑠𝑒𝑛(2𝜃) 10. 𝐸𝑠𝑏𝑜𝑧𝑎𝑟 𝑙𝑎 𝑐𝑢𝑟𝑣𝑎 𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡𝑎𝑑𝑎 𝑝𝑜𝑟 𝑙𝑎𝑠 𝑒𝑐𝑢𝑎𝑐𝑖𝑜𝑛𝑒𝑠 𝑝𝑎𝑟𝑎𝑚é𝑡𝑟𝑖𝑐𝑎𝑠 (𝑖𝑛𝑑𝑖𝑐𝑎𝑛𝑑𝑜 𝑠𝑢 𝑠𝑒𝑛𝑡𝑖𝑑𝑜) 𝑦 𝑒𝑠𝑐𝑟𝑖𝑏𝑖𝑟 𝑙𝑎 𝑒𝑐𝑢𝑎𝑐𝑖ó𝑛 𝑐𝑜𝑟𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑖𝑒𝑛𝑡𝑒 𝑒𝑙𝑖𝑚𝑖𝑛𝑎𝑛𝑑𝑜 𝑒𝑙 𝑝𝑎𝑟𝑎𝑚é𝑡𝑟𝑜 𝑥 = 3 ∗ cos(ѳ) , 𝑦 = 3 ∗ 𝑠𝑒𝑛(ѳ) 11. 𝐸𝑠𝑏𝑜𝑧𝑎𝑟 𝑙𝑎 𝑐𝑢𝑟𝑣𝑎 𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡𝑎𝑑𝑎 𝑝𝑜𝑟 𝑙𝑎𝑠 𝑒𝑐𝑢𝑎𝑐𝑖𝑜𝑛𝑒𝑠 𝑝𝑎𝑟𝑎𝑚é𝑡𝑟𝑖𝑐𝑎𝑠 (𝑖𝑛𝑑𝑖𝑐𝑎𝑛𝑑𝑜 𝑠𝑢 𝑠𝑒𝑛𝑡𝑖𝑑𝑜) 𝑦 𝑒𝑠𝑐𝑟𝑖𝑏𝑖𝑟 𝑙𝑎 𝑒𝑐𝑢𝑎𝑐𝑖ó𝑛 𝑐𝑜𝑟𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑖𝑒𝑛𝑡𝑒 𝑒𝑙𝑖𝑚𝑖𝑛𝑎𝑛𝑑𝑜 𝑒𝑙 𝑝𝑎𝑟𝑎𝑚é𝑡𝑟𝑜 𝑥 = 𝑒 3𝑡 , 𝑦 = 𝑒 𝑡 12. 𝐶𝑎𝑙𝑐𝑢𝑙𝑎𝑟 𝑒𝑙 á𝑟𝑒𝑎 𝑑𝑒 𝑙𝑎 𝑟𝑒𝑔𝑖ó𝑛 𝑟 = 2 ∗ cos(3𝜃) 13. 𝐻𝑎𝑙𝑙𝑎𝑟 𝑙𝑎 𝑙𝑜𝑛𝑔𝑖𝑡𝑢𝑑 𝑑𝑒 𝑎𝑟𝑐𝑜 𝑑𝑒 𝑙𝑎 𝑐𝑢𝑟𝑣𝑎 𝑟 = 1 + 𝑠𝑒𝑛(𝜃) 14. 𝐻𝑎𝑙𝑙𝑎𝑟 𝑒𝑙 á𝑟𝑒𝑎 𝑑𝑒 𝑙𝑎 𝑠𝑢𝑝𝑒𝑟𝑓𝑖𝑐𝑖𝑒 𝑔𝑒𝑛𝑒𝑟𝑎𝑑𝑎 𝑝𝑜𝑟 𝑙𝑎 𝑐𝑖𝑟𝑐𝑢𝑛𝑓𝑒𝑟𝑒𝑛𝑐𝑖𝑎 𝜋 𝑟 = 𝑓(𝜃) = cos(𝜃) , 𝑎𝑙 𝑔𝑖𝑟𝑎𝑟 𝑒𝑛 𝑡𝑜𝑟𝑛𝑜 𝑑𝑒 𝑙𝑎 𝑟𝑒𝑐𝑡𝑎 𝜃 = 2