Tuhinga
He ariā taketake ngā whakauru i roto i te tātaitai, e pā tata ana ki ngā pānga. Ahakoa e whakaahua ana ngā pānga i te panoni o te pikinga, te pikinga rānei o te kōpiko i tētahi pūwāhi motuhake, ko ngā whakauru, i tētahi atu ringa, e kapi ana i te horahanga katoa i raro i te kōpiko i roto i tētahi wā motuhake. He taputapu nui ngā whakauru i roto i te pāngarau, te ahupūngao, te miihini, me te maha atu o ngā mara. Ka matapakihia e tēnei tuhinga te ariā o ngā whakauru, tae atu ki tōna hītori, te whakamāramatanga, ngā momo whakauru, ngā tono, me ētahi tauira raruraru whakauru.
Hītori Whakauru
I whakaurua tuatahitia ngā whakauru e ngā kaipūtaiao Kariki onamata mā te tikanga whakangā i whakawhanakehia e Eudoxus, ā, i muri mai ka rongonuihia e Archimedes. Heoi, ko te whanaketanga o ngā whakauru hou ka taea te whai i ngā mahi a Isaac Newton rāua ko Gottfried Wilhelm Leibniz i te rautau 17. Ahakoa i mahi takitahi a Newton rāua ko Leibniz, i whakawhanakehia e rāua te tātaitai e whakauru ana i ngā whakauru me ngā taupatupatu, arā, ngā ariā e rua e hono ana i te Kaupapa Taketake o te Tātaitai. E kī ana tēnei ariā he mahi whakahurihuri te whakauru me te wehewehe.
I whakamahia e Newton ngā taunga whakauru i roto i te horopaki o te ahupūngao hei tatau i te horahanga i raro i tētahi kōpiko e tohu ana i te huringa o te tere me te wā, he ariā nui ki te miihini matarohia. I tētahi atu taha, nā Leibniz i whakawhanake te tuhi taunga whakauru ∫ e whakamahia ana e tātou i ēnei rā, i ahu mai i te reta roa "s," he whakarāpopototanga mō te summa (Latina mō te "tapeke").
Te Whakamāramatanga o te Whakauru
Ka taea te whakaaro ki te tauwehenga o tētahi mahi he "whakakore-tāpiritanga." I roto i ngā tikanga ōkawa, e rua ngā momo tauwehenga e whakamahia whānuitia ana: te tauwehenga mutunga kore me te tauwehenga tino tuturu.
Taupū Kore-Mutu
Ko te taupū mutunga kore o tētahi mahi f(x) ko te mahi F(x) ko tōna taupū o F(x) ko f(x). Ka tuhia tōna tohu penei:
∫ f(x) dx = F(x) + C
ko C he pūmau whakauru e hiahiatia ana nā te mea ko te tauwehenga o tētahi pūmau he kore. Ko tētahi tauira māmā o tētahi tauwehe mutunga kore ko:
∫ 2x dx = x² + C
Taupū Tūturu
Ko te tauwehenga tuturu o tētahi mahi f(x) mai i a ki b e whakaahua ana i te horahanga i raro i te kōpiko mai i x = a ki x = b. Ka tuhia penei:
∫[a,b] f(x) dx = F(b) – F(a)
ko F(x) te ātete-whakaputa o f(x). Hei tauira:
∫[1,3] 2x dx = (3² – 1²) = 8
Ngā Momo Whakaurunga
Haunga ngā taunga taketake kua kōrerotia i runga ake nei, kei reira anō ētahi atu momo taunga e whakamahia ana i roto i te pāngarau me ngā mahi ā-ringa. Ko ētahi o ēnei:
Taupū Takirua
Ko te whakaurunga takirua he whakaurunga ki runga i ngā taurangi e rua, neke atu rānei. He maha ngā wā ka whakamahia i roto i te tātaitanga maha-ahu hei tatau i ngā rōrahi o raro i te mata, i ngā uara tau rānei mō ngā mahi o ngā taurangi e rua, neke atu rānei. Ko ētahi tauira:
∫∫_R f(x,y) dA
Te Whakaurunga o Lebesgue
Ko te taupū Lebesgue he whakawhanuitanga o te taupū Riemann e whakamahia ana hei whakahaere i ngā mahi tino koretake. He mea tino nui tēnei taupū i roto i te tātari tūturu me te ariā tūponotanga.
Taupū Raina
Ka whakauruhia e te tauwehenga rārangi he mahi i runga i tētahi kōpiko, i tētahi ara rānei kua hoatu i roto i tētahi papa, i tētahi wāhi rānei. He tino whai hua i roto i te ahupūngao me te hangarau hei tatau i te mahi i mahia e tētahi kaha i runga i tētahi ara kua hoatu.
∫_C F · dr
Te Whakakotahitanga o te Mata
Ka whakarerekē te taunga mata i te taunga takirua kia pā ki ngā mata i roto i te wāhi toru-ahu. He maha ngā wā e whakamahia ana i roto i te hikohiko me te miihini wai.
∫∫_S f(x,y,z) dS
Taupānga Whakauru
He whānuitia ngā whakamahinga o ngā taunga whakauru i roto i ngā momo mara pūtaiao. Ko ētahi o ngā whakamahinga nui ko:
Ahupūngao
I roto i te ahupūngao, ka whakamahia ngā taunga whakauru hei tatau i te mahi, te pūngao, me ētahi atu rahinga ā-tinana. Hei tauira, ka whakamahia ngā taunga whakauru hei tatau i te mahi i mahia e te kaha i roto i te nekehanga o te matūriki:
W = ∫ F(x) dx
He mea nui anō hoki ngā taunga whakauru mō te kimi i te papatipu, te pokapū o te papatipu, te wā o te korehau, me te tohatoha papatipu i roto i ngā mea ā-tinana.
Tauanga
I roto i ngā tatauranga me te ariā tūponotanga, ka whakamahia ngā taunga whakauru hei tatau i ngā tūponotanga me ngā tumanakohanga o ngā momo tohatoha tūponotanga. Hei tauira, ko te tatau i te tūponotanga tāpiripiri o tētahi taurangi matapōkere tonu:
P(X ≤ x) = ∫_(-∞, x) f(t) dt
ōhanga
I roto i te ōhanga, ka whakamahia ngā taunga whakauru hei tatau i ngā hua katoa, ngā utu, me te toenga o te kaihoko/kaihanga. Hei tauira, hei tatau i ngā utu katoa i roto i te horopaki o te utu taha:
Utu Katoa = ∫ MC(q) dq
hangarau
I roto i te hangarau, he maha ngā wā ka whakamahia ngā whakauru i roto i te tātari ahotea, te whakarerekētanga āhua, me te whakawhiti wera. Ka whakamahia hoki e te hangarau whakahaere ngā whakauru mō ngā pūnaha whakahaere whakauru i roto i ngā kaiwhakahaere PID (Proportional-Integral-Derivative).
Ngā Tauira Raru Whakauru
Anei ētahi tauira o ngā raruraru whakauru me ā rātou otinga:
Tauira 1: Taurite Kore-Mutu
Kimihia te taupū mutunga kore o f(x) = 3x²:
∫ 3x² dx
Rongoā:
Ka taea te whakaoti i tēnei taupū mā te whakamahi i te ture taupū taketake. Ka tāpirihia he kotahi ki te taupū, kātahi ka wehea ki te taupū hou:
∫ 3x² dx = 3 (1/3)x³ = x³ + C
Tauira 2: Taurite Tūturu
Kimihia te horahanga i raro i te kōpiko f(x) = 2x mai i te 1 ki te 4:
∫[1,4] 2x dx
Rongoā:
Tuatahi, ka kitea e tātou te ārai-whakaputa o 2x, arā, ko x². Kātahi ka whakamahia ngā rohenga o raro me o runga:
∫[1,4] 2x dx = [x²]₁^₄ = 4² – 1² = 16 – 1 = 15
Whakamutunga
He ariā nui ngā taupū i roto i te tātaitai, e hopu ana i te ariā o te horahanga i raro i te kōpiko me ōna whakamahinga maha. Mai i tō rātou hītori i Kariki onamata ki ngā whanaketanga nui a Newton rāua ko Leibniz, ki ō rātou momo me ngā whakamahinga i roto i te pūtaiao me te miihini, he mahi nui tā ngā taupū i roto i te anga pāngarau o ēnei rā. Mā te mārama ki ngā taupū ehara i te mea ka āwhina i a tātou ki te whakaoti rapanga pāngarau anake, engari ka whakatuwhera hoki i ngā māramatanga ki ō rātou whakamahinga maha i roto i te oranga o ia rā me te ao ngaio.
