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 + [