Ngā Āhuatanga o ngā Taurite Tūturu: Ngā Whakamahinga me ngā Ariā Taketake
Pendahuluan
Ko ngā taupū tetahi o ngā ariā tino taketake o te tātaitai, me ngā taupatupatu. He maha ngā whakamahinga o ngā taupū tuturu i roto i te pūtaiao, te hangarau, me te ōhanga. Ka hoatu e te taupū tuturu o tētahi mahi he uara e pā ana ki te horahanga i raro i te kōpiko o taua mahi i roto i tētahi wā kua whakaritea. Ka whakamāramahia e tēnei tuhinga ētahi āhuatanga taketake o ngā taupū tuturu, ka whakaratohia he tauira whakamahinga, ā, ka tūhuratia ngā pānga mahi o ia āhuatanga.
He Kupu Whakataki ki ngā Whakaurunga Tūturu
Hei tīmatanga ki te mārama ki ngā taunga whakapūmau, me tautuhi tātou he aha te taunga whakapūmau. Me kī ko \( f(x) \) he mahi tonu i runga i te wā \([a, b]\). Ko te taunga whakapūmau o \( f(x) \) mai i \( a \) ki \( b \) ka tohua e:
\[ \int_{a}^{b} f(x) \, dx \]
Mā tēnei uara ka puta te horahanga kua tatauhia i raro i te kōpiko \( f(x) \) mai i \( x = a \) ki \( x = b \).
Ngā Āhuatanga o ngā Taurite Tūturu
1. Te Raina
He āhuatanga o te rārangi ngā taupūnga tuturu, arā, ko te taupūnga o te tapeke o ngā mahi maha he ōrite ki te tapeke o ngā taupūnga o ia mahi. I te nuinga o te wā, mēnā he mahi tonu a \( f(x) \) me \( g(x) \) i runga i \([a, b]\) ā, he pūmau a \( c \), kāti:
\[ \int_{a}^{b} [cf(x)] \, dx = c \int_{a}^{b} f(x) \, dx \]
\[ \int_{a}^{b} [f(x) + g(x)] \, dx = \int_{a}^{b} f(x) \, dx + \int_{a}^{b} g(x) \, dx \]
Ko tētahi tauira o te whakamahinga o tēnei āhuatanga rārangitanga ko te wā e hiahia ana tātou ki te tatau i te horahanga i raro i te kōpiko o tētahi mahi matatini ka taea te wehewehe ki ētahi mahi māmā ake.
2. Tāpiritanga (Tāpiritanga Wā)
Ko te āhuatanga nui e whai ake nei ko te āhuatanga tāpiri, e kī ana ko te tauwehe i runga i te hononga o ngā āputa tata ko te tapeke o ngā tauwehe i runga i ia o aua āputa. Mena \( a < c < b \), kāti: \[ \int_{a}^{b} f(x) \, dx = \int_{a}^{c} f(x) \, dx + \int_{c}^{b} f(x) \, dx \] He whai hua tēnei āhuatanga ina hiahia tātou ki te tatau i tētahi taupū i runga i tētahi āputa nui mā te wehewehe i a ia ki ngā āputa iti ake, he ngāwari ake te tatau. 3. Whānui Kore (Whānui Kore) Mēnā ka whakauruhia e tātou tētahi mahi ki tētahi āputa he whānui kore tōna, ko te hua o te taupū he kore. Pāngarau: \[ \int_{a}^{a} f(x) \, dx = 0 \] He āhuatanga ngāwari tēnei, nā te mea ko te horahanga i raro i te kōpiko i runga i te āputa kore-ahu he kore. 4. Te Hurihanga o ngā Here Mā te whakarerekē i te raupapa o ngā rohe o tētahi taupū ka huri te tohu o te taupū: \[ \int_{a}^{b} f(x) \, dx = -\int_{b}^{a} f(x) \, dx \] He whai hua tēnei i roto i ngā āhuatanga maha, inā koa ka hiahiatia te whakarerekētanga tohu hei tatau i te uara o te taupū. 5. He āhuatanga whakatairite anō hoki tō ngā Whakakotahitanga Whakatairite. Mena e rua ngā mahi \( f(x) \) me \( g(x) \) e haere tonu ana i runga i \([a, b]\) me \( f(x) \leq g(x) \) mō ngā \( x \) katoa i roto i \([a, b]\), kāti: \[ \int_{a}^{b} f(x) \, dx \leq \int_{a}^{b} g(x) \, dx \] He mea nui tēnei āhuatanga i roto i te tātari i ngā uara whakauru mō te whakatata me ngā tikanga tau. 6. Te Ariā Uara Toharite mō ngā Whakaurunga Mena he pumau te \( f(x) \) i runga i te \([a, b]\), kāti he \( c \) i roto i te \([a, b]\) kia penei: \[ \int_{a}^{b} f(x) \, dx = f(c) \cdot (b-a) \] Ko te tikanga he uara toharite o \( f(x) \) i runga i te āputa, ā, mā te whakarea i te whānui o te āputa ka puta te uara o te whakaurunga. 7. Te Kaupapa Taketake o te Tātaitai Ka honoa e tēnei ariā te ariā o te taupū mutunga ki te taupū, ka wehea kia rua ngā wāhanga: - Wāhanga Tuatahi: Mena he pumau a \( f \) i runga i te \([a, b]\) ā, ko \( F \) he taupū-kore o \( f \) (arā, \( F' = f \)), kātahi: \[ \int_{a}^{b} f(x) \, dx = F(b) - F(a) \] - Wāhanga Tuarua: Mena he mahi pumau a \( f \) i runga i te wā \([a, b]\) ā, ko te tautuhi o \( G \) e: \[ G(x) = \int_{a}^{x} f(t) \, dt \] kātahi ka pumau a \( G \) i runga i te \([a, b]\), te rerekētanga i runga i te wā tuwhera \((a, b)\), ā, ko \( G'(x) = f(x) \). Te Whakamahi i ngā Āhuatanga o ngā Taupoki Tūturu Mā te whakamahi i ngā āhuatanga o ngā taupoki tūturu i roto i ngā tātaitanga mahi ka taea e tātou te whakahaere i ngā rapanga uaua ki ngā rapanga ngāwari ake. Anei ētahi tauira o ngā tono: Te Tatau i te Horahanga Ko te tatau i te horahanga i raro i tētahi kōpiko he maha ngā wā e hiahiatia ana te wehewehe i tētahi āputa matatini ki ngā wāhanga iti ake, me te whakamahi i te rārangitanga me te āhuatanga tāpiri: \[ \text{Horahanga} = \int_{a}^{c} f(x) \, dx + \int_{c}^{b} f(x) \, dx \] Ahupūngao: Te Mahi me te Pūngao I roto i te ahupūngao, ka whakamahia ngā taunga whakauru hei tatau i te mahi i mahia e tētahi kaha taurangi. Mena ko te kaha hei mahi a te tūranga, ko te mahi i mahia mai i te tūranga \( x = a \) ki \( x = b \) ko: \[ W = \int_{a}^{b} F(x) \, dx \] Ōhanga: Tapeke Moni Whiwhi I roto i te ōhanga, mena ko te \( p(x) \) he mahi a te utu mō ia rahinga waeine o te taonga i hokona, ko te tapeke moni whiwhi mai i te maha o ngā waeine \( a \) ki \( b \) o te taonga i hokona ko: \[ \text{Tapeke Moni Whiwhi} = \int_{a}^{b} p(x) \, dx \] Whakamutunga He taputapu tino nui te tauwehenga tino i roto i te pāngarau tono, ā, he maha ngā āhuatanga whai hua e taea ai e tātou te whakahaere me te whakaoti rapanga uaua. Ko ngā āhuatanga pēnei i te rārangitanga, te tāpiritanga, me te ariā taketake o te tātaitai he tūāpapa pakari mō ngā tataunga me ngā tātaritanga pāngarau atu anō. Mā te mārama me te whakamahi tika i ēnei āhuatanga ka taea e tātou te whakaoti rapanga i roto i te whānuitanga o ngā mara, mai i te ahupūngao ki te ōhanga.
