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

𝑑𝑥=

𝑑𝑦𝑑𝑡𝑑𝑥𝑑𝑡

, 𝑐𝑜𝑛 𝑑𝑥

𝑑𝑡≠ 0

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:

𝐿 = ∫ √(𝑑𝑥

𝑑𝑡)

2

+ (𝑑𝑦

𝑑𝑡)

2𝑏

𝑎

𝑑𝑡

∗ 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:

𝑆 = 2π ∫ 𝑔(𝑡)√(𝑑𝑥

𝑑𝑡)

2

+ (𝑑𝑦

𝑑𝑡)

2𝑏

𝑎

𝑑𝑡 𝑒𝑗𝑒 𝑑𝑒 𝑟𝑒𝑣𝑜𝑙𝑢𝑐𝑖ó𝑛 𝑥; 𝑔(𝑡) ≥ 0

𝑆 = 2π ∫ 𝑓(𝑡)√(𝑑𝑥

𝑑𝑡)

2

+ (𝑑𝑦

𝑑𝑡)

2𝑏

𝑎

𝑑𝑡 𝑒𝑗𝑒 𝑑𝑒 𝑟𝑒𝑣𝑜𝑙𝑢𝑐𝑖ó𝑛 𝑦; 𝑓(𝑡) ≥ 0

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)

𝑡𝑔 (ѳ) =𝑦

𝑥 𝑟2 = 𝑥2 + 𝑦2

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

𝑑𝑥=

𝑑𝑦𝑑ѳ𝑑𝑥𝑑ѳ

, 𝑐𝑜𝑛 𝑑𝑥

𝑑ѳ≠ 0 𝑒𝑛 (𝑟, ѳ)

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:

𝐴 =1

2∫ [𝑓(ѳ)]2 𝑑ѳ =

1

2

𝛽

𝛼

∫ 𝑟2 𝑑ѳ𝛽

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:

𝑠 = ∫ √[𝑓(𝜃)]2 + [𝑓´(𝜃)]2 𝑑𝜃 =𝛽

𝛼

∫ √𝑟2 + [𝑑𝑟

𝑑𝜃]

2

𝑑𝜃𝛽

𝛼

∗ 𝑨𝒓𝒆𝒂 𝒅𝒆 𝒖𝒏𝒂 𝒔𝒖𝒑𝒆𝒓𝒇𝒊𝒄𝒊𝒆 𝒅𝒆 𝒓𝒆𝒗𝒐𝒍𝒖𝒄𝒊ó𝒏

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:

𝑆 = 2𝜋 ∫ 𝑓(𝜃)𝑠𝑒𝑛𝜃√[𝑓(𝜃)]2 + [𝑓´(𝜃)]2 𝑑𝜃 𝑒𝑛 𝑡𝑜𝑟𝑛𝑜 𝑎𝑙 𝑒𝑗𝑒 𝑝𝑜𝑙𝑎𝑟𝛽

𝛼

𝑆 = 2𝜋 ∫ 𝑓(𝜃)𝑐𝑜𝑠𝜃√[𝑓(𝜃)]2 + [𝑓´(𝜃)]2 𝑑𝜃 𝑒𝑛 𝑡𝑜𝑟𝑛𝑜 𝑎 𝑙𝑎 𝑟𝑒𝑐𝑡𝑎 𝜃 =𝜋

2

𝛽

𝛼

𝑇𝑎𝑙𝑙𝑒𝑟 𝐶á𝑙𝑐𝑢𝑙𝑜 𝑀𝑢𝑙𝑡𝑖𝑣𝑎𝑟𝑖𝑎𝑑𝑜

𝐺𝑟𝑎𝑓𝑖𝑐𝑎𝑟

1. 𝑟(ѳ) = 3

2. ѳ =𝜋

6

3. 𝑟(ѳ) = 3ѳ

4. 𝑟(ѳ) = 0.5 + cos(ѳ)

5. 𝑟(ѳ) =3

2+ 2 ∗ cos(ѳ)

6. 𝑟(ѳ) = 2 ∗ 𝑠𝑒𝑐(ѳ) + 3

7. 𝑟(ѳ) =5

2 − 2𝑠𝑒𝑛(ѳ)

8. 𝑟(ѳ) = 16 ∗ cos(2ѳ)

9. 𝑟(ѳ) = 1 + 3 ∗ 𝑠𝑒𝑛 (ѳ

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 = 𝑒3𝑡, 𝑦 = 𝑒𝑡

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

𝑟 = 𝑓(𝜃) = cos(𝜃) , 𝑎𝑙 𝑔𝑖𝑟𝑎𝑟 𝑒𝑛 𝑡𝑜𝑟𝑛𝑜 𝑑𝑒 𝑙𝑎 𝑟𝑒𝑐𝑡𝑎 𝜃 =𝜋

2