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 taupū i runga i te huinga o ngā āputa tata ko te tapeke o ngā taupū i runga i ia 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 ki ngā āputa iti ake, ngāwari ake te tatau. 3. Whānui Kore Mena ka whakauruhia e tātou tētahi mahi i runga i tētahi āputa he whānui kore, ko te hua o te taupū he kore. I roto i te pāngarau: \[ \int_{a}^{a} f(x) \, dx = 0 \] He āhuatanga ngāwari tēnei, nā te mea ko te horahanga i raro i te pihi i runga i tētahi āputa kore-ahu he kore. 4. Te Whakahuri i ngā Here (Pembalik Batas) Mā te whakarerekē i te raupapa o ngā rohe o tētahi taupū ka whakarerekē i 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. Te Whakataurite (Perbandingan)
He āhuatanga whakataurite anō hoki tō ngā taupūnga taupū. Mena he mahi tonu ngā mahi e rua \( f(x) \) me \( g(x) \) 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 taupūnga 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 Ariā Taketake o te Tātaitai (Te Ariā Taketake o te Tātaitai) Ka hono tēnei ariā i 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 \([a, b]\), he rerekētanga i runga i te wā tuwhera \((a, b)\), ā, ko \( G'(x) = f(x) \). Te Whakamahinga o ngā Āhuatanga o ngā Taupūnga Tūturu Mā te whakamahi i ngā āhuatanga o ngā taupūnga tūturu i roto i ngā tātaitanga mahi ka taea e tātou te whakahaere i ngā raruraru uaua ki ngā mea ngāwari ake te whakahaere. Anei ētahi tauira o ngā whakamahinga: Te Tatau i te Horahanga Ko te tatau i te horahanga i raro i te kōpiko he maha ngā wā e hiahiatia ana te wehewehe i te wā uaua ki ngā wāhanga iti ake me te whakamahi i te rārangi 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 taupūnga tūturu hei tātai i te mahi i mahia e te 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 āhei ai tātou ki te whakahaere me te whakaoti rapanga uaua. Ko ngā āhuatanga pēnei i te rārangi, te tāpiritanga, me te ariā taketake o te tātaitai he turanga pakari mō ngā tataunga pāngarau me ngā tātaritanga. Mā te mārama me te whakamahi pai 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.