Appunti Metodi Matematici e Statistici – Scheda Esercitazione N. 5 – Integrali Definiti

Ancona, 6 dicembre 2011 1

DETERMINARE I SEGUENTI INTEGRALI DEFINITI

1. 𝑥

1+ 𝑥4

1

0𝑑𝑥 =

𝜋

8

2. 𝑙𝑛𝑥

𝑥

𝑒

1𝑑𝑥 = 2 (2 − 𝑒)

3. 𝑒𝑥− 1

𝑒𝑥 + 1

1

0𝑑𝑥 = 2𝑙𝑛

𝑒 +1

2− 1

4. 𝑥2 − 3𝑥 4

−1𝑑𝑥 =

49

6

5. 𝑎𝑟𝑐𝑡𝑔 3 𝑥

2−𝑥

1

−1𝑑𝑥 =

𝜋

3−

3

4𝑙𝑛3

6. 𝑥

𝑥2− 2𝑥−3

0

−1𝑑𝑥 =

3

4𝑙𝑛3 − 𝑙𝑛2

CALCOLARE LE SEGUENTI AREE/VOLUMI

1. 𝐴𝑟𝑒𝑎 𝑑𝑒𝑙𝑙𝑎 𝑠𝑢𝑝𝑒𝑟𝑓𝑖𝑐𝑖𝑒 𝑝𝑖𝑎𝑛𝑎 𝑑𝑒𝑙𝑖𝑚𝑖𝑡𝑎𝑡𝑎 𝑑𝑎 𝑓 𝑥 = 𝑠𝑖𝑛𝑥 𝑒 𝑙′𝑎𝑠𝑠𝑒 𝑑𝑒𝑙𝑙𝑒 𝑥 𝑐𝑜𝑛 𝑥 ∈ 0, 𝜋

𝑨𝒓𝒆𝒂 = 𝟐

2. 𝐴𝑟𝑒𝑎 𝑑𝑒𝑙𝑙𝑎 𝑠𝑢𝑝𝑒𝑟𝑓𝑖𝑐𝑖𝑒 𝑝𝑖𝑎𝑛𝑎 𝑑𝑒𝑙𝑖𝑚𝑖𝑡𝑎𝑡𝑎 𝑑𝑎𝑙𝑙𝑒 𝑑𝑢𝑒 𝑝𝑎𝑟𝑎𝑏𝑜𝑙𝑒 𝑦2 = 4𝑥 𝑒 𝑥2 = 4𝑦

𝑨𝒓𝒆𝒂 =𝟏𝟔

𝟑

3. 𝐴𝑟𝑒𝑎 𝑑𝑒𝑙𝑙𝑎 𝑠𝑢𝑝𝑒𝑟𝑓𝑖𝑐𝑖𝑒 𝑝𝑖𝑎𝑛𝑎 𝑑𝑒𝑙𝑖𝑚𝑖𝑡𝑎𝑡𝑎 𝑑𝑎𝑙𝑙𝑒 𝑝𝑎𝑟𝑎𝑏𝑜𝑙𝑒 𝑦 = 𝑥2 − 3𝑥 + 2 𝑒

𝑦 = −𝑥2 + 𝑥 − 2

𝑨𝒓𝒆𝒂 =𝟖

𝟑

4. 𝐴𝑟𝑒𝑎 𝑑𝑒𝑙𝑙𝑎 𝑠𝑢𝑝𝑒𝑟𝑓𝑖𝑐𝑖𝑒 𝑝𝑖𝑎𝑛𝑎 𝑑𝑒𝑙𝑖𝑚𝑖𝑡𝑎𝑡𝑎 𝑑𝑎𝑙𝑙𝑎 𝑝𝑎𝑟𝑎𝑏𝑜𝑙𝑒 𝑦2 = 4𝑥 𝑒 𝑙𝑎 𝑟𝑒𝑡𝑡𝑎

2𝑥 + 𝑦 − 4 = 0 𝑒 𝑙′𝑎𝑠𝑠𝑒 𝑑𝑒𝑙𝑙𝑒 𝑥

𝑨𝒓𝒆𝒂 =𝟕

𝟑

5. 𝑉𝑜𝑙𝑢𝑚𝑒 𝑑𝑒𝑙 𝑠𝑜𝑙𝑖𝑑𝑜 𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑜 𝑑𝑎𝑙𝑙𝑎 𝑟𝑜𝑡𝑎𝑧𝑖𝑜𝑛𝑒 𝑎𝑡𝑡𝑜𝑟𝑛𝑜 𝑎𝑙𝑙′𝑎𝑠𝑠𝑒𝑑𝑒𝑙𝑙𝑒 𝑥 𝑑𝑒𝑙𝑙𝑎 𝑝𝑜𝑟𝑧𝑖𝑜𝑛𝑒 𝑑𝑖 𝑝𝑖𝑎𝑛𝑜

𝑙𝑖𝑚𝑖𝑡𝑎𝑡𝑎 𝑑𝑎𝑙𝑙𝑎 𝑐𝑢𝑟𝑣𝑎 𝑦 = 2 𝑥 𝑒 𝑑𝑎𝑙𝑙𝑒 𝑟𝑒𝑡𝑡𝑒 𝑥 = 1 𝑒 𝑥 = 3 𝑒 𝑑𝑎𝑙𝑙′𝑎𝑠𝑠𝑒 𝑑𝑒𝑙𝑙𝑒 𝑥

𝑽𝒐𝒍𝒖𝒎𝒆 = 𝟏𝟔𝝅

6. 𝑉𝑜𝑙𝑢𝑚𝑒 𝑑𝑒𝑙 𝑠𝑜𝑙𝑖𝑑𝑜 𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑜 𝑑𝑎𝑙𝑙𝑎 𝑟𝑜𝑡𝑎𝑧𝑖𝑜𝑛𝑒 𝑎𝑡𝑡𝑜𝑟𝑛𝑜 𝑎𝑙𝑙′𝑎𝑠𝑠𝑒𝑑𝑒𝑙𝑙𝑒 𝑥 𝑑𝑒𝑙𝑙𝑎 𝑝𝑜𝑟𝑧𝑖𝑜𝑛𝑒 𝑑𝑖 𝑝𝑖𝑎𝑛𝑜

𝑙𝑖𝑚𝑖𝑡𝑎𝑡𝑎 𝑑𝑎𝑙𝑙𝑎 𝑐𝑢𝑟𝑣𝑎 𝑦 = 1

1 + 4𝑥2 𝑒 𝑑𝑎𝑙𝑙𝑒 𝑟𝑒𝑡𝑡𝑒 𝑥 =

1

2 𝑒 𝑥 =

3

2 𝑒 𝑑𝑎𝑙𝑙′𝑎𝑠𝑠𝑒 𝑑𝑒𝑙𝑙𝑒 𝑥

𝑽𝒐𝒍𝒖𝒎𝒆 =𝝅

𝟐𝟒