Ngā Tauira o ngā Taupānga Whakauru i te Oranga o Ia Rā
He ariā taketake te whakaurunga i roto i te tātaitai, me ngā tono kanorau i roto i ngā momo mara o te pūtaiao me te oranga o ia rā. Ko te whakaurunga te tukanga kimi i ngā taunga whakauru, ka taea te tautuhi hei te tapeke o ngā taunga iti rawa, hei kimi rānei i te horahanga i raro i tētahi kōpiko kua hoatu. Ahakoa he maha ngā wā e kiia ana te ariā o te whakaurunga he mea whakarāpopoto me te ariā, he maha ngā raruraru mahi ka taea te whakaoti mā te whakamahi i ngā taunga whakauru. Ka matapakihia e tēnei tuhinga ētahi tauira o ngā tono taunga whakauru i roto i te oranga o ia rā.
1. Te Tātaitanga o te Horahanga me te Rōrahi
Ko tētahi o ngā whakamahinga tino noa o ngā taupū ko te tatau i te horahanga me te rōrahi. I roto i te āhuahanga, ka whakamahia ngā taupū hei tatau i te horahanga mata o ngā mea kāore he āhuahanga māmā.
a. Te horahanga i raro i te kōpiko
Hei whakatau i te horahanga i raro i te kōpiko, ka taea e tātou te whakamahi i ngā tauwehenga. Hei tauira, hei kimi i te horahanga i raro i te kōpiko kauwhata mahi f(x) mai i a ki b, ka taea e tātou te tuhi:
\[ \text{Horahanga} = \int_{a}^{b} f(x) \, dx \]
b. Te Rōrahi o ngā Mea Hurihuri
Ka taea hoki te tatau i te rōrahi o tētahi totoka i hangaia mā te huri i te rohe i raro i tētahi kōpiko huri noa i tētahi tuaka kua hoatu mā te whakamahi i ngā taunga whakauru. Ko te tikanga kōpae me te tikanga mowhiti he tikanga e rua e whakamahia whānuitia ana. Hei tauira, ko te rōrahi o tētahi totoka i hangaia mā te huri i te kōpiko y = f(x) mai i x = a ki x = b huri noa i te tuaka-x ka taea te tatau penei:
\[ V = \pi \int_{a}^{b} [f(x)]^2 \, dx \]
2. Ahupūngao me te Hangarau
He maha ngā ariā o te ahupūngao me te hangarau e whakamahi ana i ngā tauwehenga hei whakatauira i ngā āhuatanga taiao.
a. Te Tātai i te Mahi
Ka taea te tatau i te mahi i mahia e te kaha i roto i tētahi nekehanga kua hoatu mā te whakamahi i tētahi taupū. Hei tauira, ki te rerekē te kaha F(x) i te ara mai i x = a ki x = b, ko te mahi i mahia ko:
\[ W = \int_{a}^{b} F(x) \, dx \]
b. Te Tātai i te Wā Āhuakore
Ko te wā o te korekore he mehua o te tohatoha o te papatipu o tētahi mea e pā ana ki tōna tuaka hurihuri. Mō tētahi mea tonu, ka taea te tatau i te wā o te korekore I penei:
\[ I = \int r^2 \, dm \]
ko r te tawhiti i waenganui i te huānga papatipu dm me te tuaka hurihuri.
c. Te Tohatoha Utaina
I roto i te hiko pūmau, ka whakamahia ngā taunga whakauru hei tatau i te mara hiko me te pūmanawa hiko mai i tētahi tohatoha utu tonu. Hei tauira, hei kimi i te pūmanawa V i tētahi pūwāhi kua hoatu nā te tohatoha utu, ka taea e tātou te whakamahi i te taunga whakauru:
\[ V = \int \frac{k \, dq}{r} \]
ko k te pūmau o Coulomb, ko dq te huānga utu, ā, ko r te tawhiti i waenganui i te huānga utu me te pūwāhi tirotiro.
3. Ōhanga
I te ao ohaoha, e whakamahia ana te ariā o te tauwehenga mō te tātari pūtea me te whakahaere mōrearea.
a. Te Mahi Tohatoha Tūponotanga
He maha ngā wā ka whakamahia ngā taunga whakauru hei kimi i te mahi tohatoha tāpiripiri (CDF) o tētahi taurangi matapōkere. Hei tauira, mēnā ko f(x) te mahi kiato tūponotanga (PDF) o tētahi taurangi matapōkere X, ka taea te tatau i te CDF F(x) penei:
\[ F(x) = \int_{-\infty}^{x} f(t) \, dt \]
b. Te toenga o te kaihoko me te kaihanga
Ko te toenga kaihoko te rerekētanga i waenga i te utu e pai ana ngā kaihoko ki te utu me te utu e utua ana e rātou. Waihoki, ko te toenga kaihanga te rerekētanga i waenga i te utu ka riro i a rātou me te utu iti rawa e pai ana rātou ki te whakaae. Ka taea te tatau i ēnei ariā e rua mā te whakamahi i ngā taunga whakauru i runga i ngā ānau tono me te ānau tuku.
\[ \text{Te Tapeke Kaihoko} = \int_{0}^{Q} (D(q) – P) \, dq \]
\[ \text{Te Tapeke o te Kaihanga} = \int_{0}^{Q} (P – S(q)) \, dq \]
ko D(q) te mahi tono, ko S(q) te mahi tuku, ko P te utu taurite, ā, ko Q te rahinga taurite.
4. Koiora me te Rongoā
He whānuitia ngā tono o ngā whakauru i roto i te koiora me te rongoā, inā koa i roto i ngā tauira pāngarau me te tātari raraunga.
a. Te tipu o te taupori
He maha ngā wā ka uru atu ngā tauira tipu taupori ki ngā whārite rerekētanga ka taea te whiwhi i ngā otinga mā te whakauru. Hei tauira, i roto i te tauira tipu taupū, ko te tere o te huringa o te taupori P(t) e pā ana ki te taupori i roto i te wā \( t \) mā te whārite rerekētanga:
\[ \frac{dP}{dt} = rP \]
ko r te tere tipu. Ko te otinga whakauru o tēnei whārite ka homai:
\[ P(t) = P(0)e^{rt} \]
b. Ngā mahi rongoā
E ako ana te Pharmacokinetics i te tukatuka o ngā rongoā i roto i te tinana. Ka whakamahia ngā integrals hei whakatau i te kukū o te rongoā i roto i te toto i tētahi wā motuhake, i runga i te tere o te whakahaere me te whakakorenga o te rongoā. Hei tauira, ka taea te kimi i te tapeke o te rongoā i roto i te tinana i tētahi wā mā te integrals o te tere o te huringa o te kukū o te rongoā:
\[ A(t) = \int_{0}^{t} C(t) \, dt \]
5. Ngā Tatauranga me te Tātari Raraunga
He taputapu nui ngā taupū i roto i ngā tatauranga me te tātari raraunga, inā koa i roto i te tatau i ngā tūponotanga, ngā tumanakohanga, me ngā tohatoha.
a. Te Tumanako Pāngarau
Ka taea te tatau i te tumanako pāngarau o tētahi taurangi matapōkere tonu X me te mahi kiato f(x) mā te whakamahi i te taupū:
\[ E(X) = \int_{-\infty}^{\infty} xf(x) \, dx \]
b. Te tūponotanga
Ka whakamahia ngā taupū whakauru hei tatau i te tūponotanga o te puta o tētahi taurangi matapōkere i roto i tētahi awhe kua hoatu. Hei tauira, ko te tūponotanga kei waenganui i a me b tētahi taurangi matapōkere X:
P(a X b) = f(x)
Te Katinga
Ko ngā taupū he ariā pāngarau e whai wāhi nui ana ki ngā mahi maha o ia rā. Mai i te tatau i te horahanga me te rōrahi, me ngā tono i roto i te ahupūngao me te miihini, ki te ōhanga, te koiora, me ngā tatauranga, ka āwhina ngā taupū i a tātou ki te whakatauira, ki te tātari, me te whakaoti rapanga tino uaua. He pūkenga nui te kaha ki te whakamahi pai i ngā taupū, i roto i te pūtaiao me ngā tono mahi o ia rā.
