Te Whakapūmautanga Tūturu: Te Whakamārama, Te Ariā, me te Whakamahinga
Ko te whakauru tētahi o ngā ariā matua o te tātaitai e whai wāhi nui ana ki ngā momo mara pūtaiao, tae atu ki te pāngarau, te ahupūngao, te miihini, me te ōhanga. Ko te whakauru tino tika he momo whakauru he rohe whakauru motuhake, arā, he rohe raro me te rohe runga, e tohu ana i te wā whakauru. Kāore i rite ki ngā whakauru mutunga kore e whakaputa ana i ngā mahi ārai-whakaputa, he uara tau ngā whakauru tino tika, ā, he maha ngā wā ka whakamahia hei tatau i te horahanga i raro i te pihi, te rōrahi o ngā totoka hurihuri, me ētahi atu tono mahi.
Te Whakamāramatanga o te Taupū Tūturu
Ko te taupū tuturu o tētahi mahi \( f(x) \) i runga i te wā \([a, b]\) ka tohua penei:
\[ \int_{a}^{b} f(x) \, dx \]
I konei, ko \( a \) me \( b \) ngā rohe o raro me te rohe o runga o te whakaurunga. Mā tēnei whakaurunga ka puta he tau e tohu ana i te kohikohinga o ngā uara o te mahi \( f(x) \) i roto i te awhe \( a \) ki \( b \). Mā te āhuahanga, ka taea te tautuhi i tētahi taunga tino rite ki te horahanga e herea ana e te pihi \( y = f(x) \), te tuaka-x, me ngā rārangi poutū \( x = a \) me \( x = b \).
Te Ariā Taketake o te Whakapūmautanga Tūturu
Ngā Kaupapa Taketake o te Tātaitai
E hono ana te ariā taketake o te Tātaitai i te ariā o ngā taunga whakauru ki te ariā o ngā pānga (whakarerekētanga). E rua ngā wāhanga o tēnei ariā:
1. Wāhanga Tuatahi o te Ariā: Mena he antiderivative (mahi taketake) a \( F \) o te mahi \( f \) i runga i te wā \([a, b]\), kāti:
\[ \int_{a}^{b} f(x) \, dx = F(b) – F(a) \]
E whakaatu ana tēnei wāhanga ka taea te tatau i te taupū tūturu mā te kimi i te ātete-whakaputa o \( f(x) \), kātahi ka tatau i te rerekētanga i waenga i ngā uara o te ātete-whakaputa i ngā rohe o runga me raro.
2. Wāhanga Tuarua o te Ariā: Mena he mahi tonu a \( f \) i runga i a \([a, b]\) ā, he mahi a \( F(x) \) kua tautuhia penei:
\[ F(x) = \int_{a}^{x} f(t) \, dt \]
kātahi ka \( F'(x) = f(x) \). E whakaatu ana tēnei ko te tauwehenga o te taupū o tētahi mahi he ōrite ki te mahi tonu.
Tikanga Tātai
Ko te tatau tātari o ngā taupūnga tuturu e rua ngā taahiraa matua:
– Kimihia te ātete-tāpiritanga \( F(x) \) o te mahi kua hoatu \( f(x) \).
– Tātaihia te uara o \( F \) i ngā rohe o runga me raro o te whakaurunga, kātahi ka kimihia te rerekētanga hei whiwhi i te hua whakauru.
Hei tauira, me kī tātou e hiahia ana ki te tatau i te \( \int_{2}^{5} 3x^2 \, dx \).
1. Ko te tauwehenga ā-whakaheke o \( 3x^2 \) ko \( F(x) = x^3 \).
2. Tātaihia \( F \) i ngā rohe o runga me raro:
\[ F(5) = 5^3 = 125 \]
\[ F(2) = 2^3 = 8 \]
Nō reira, \[ \int_{2}^{5} 3x^2 \, dx = 125 – 8 = 117 \]
Ngā Taupānga Whakauru Tūturu
Te Horahanga i raro i te Ārai
Ko tētahi o ngā whakamahinga tino noa o te taupū tino ko te tatau i te horahanga i raro i tētahi kōpiko. Me kī tātou e hiahia ana ki te tatau i te horahanga i raro i te kōpiko \( y = f(x) \) mai i \( x = a \) ki \( x = b \). Ka taea e tātou te whakamahi i te taupū tino hei kimi i tēnei horahanga:
\[ \text{Horahanga} = \int_{a}^{b} f(x) \, dx \]
Te Rōrahi o ngā Mea Hurihuri
Ka taea hoki te whakamahi i ngā taunga taupū hei tatau i te rōrahi o ngā mea e puta mai ana i te hurihanga o tētahi kōpiko huri noa i te tuaka-x, i te tuaka-y rānei. Ko ngā tikanga e whakamahia whānuitia ana ko te tikanga kōpae me te tikanga anga-puoto.
Tikanga Kōpae
Me kī he kōpiko tā tātou \( y = f(x) \) ā, e hiahia ana tātou ki te huri i tēnei kōpiko huri noa i te tuaka-x mai i \( x = a \) ki \( x = b \). Ka taea te tatau i te rōrahi o te mea hua mā te whakamahi i tētahi tauwehenga tino rite:
\[ V = \pi \int_{a}^{b} [f(x)]^2 \, dx \]
Tikanga Kiri Ngongo
Ki te hiahia tātou ki te huri i te kōpiko \( x = g(y) \) huri noa i te tuaka-y mai i \( y = c \) ki \( y = d \), ka taea te tatau i tōna rōrahi mā te whakamahi i:
\[ V = 2\pi \int_{c}^{d} y \, g(y) \, dy \]
Ētahi atu Taupānga
I roto i te ahupūngao, ka whakamahia ngā taunga whakapūmau hei tatau i ngā rahinga rerekē pērā i te mahi i mahia e te kaha \( F(x) \) i runga i te tawhiti \( x \), e whakaatuhia ana penei:
\[ W = \int_{a}^{b} F(x) \, dx \]
I roto i te ōhanga, ka taea te whakamahi i ngā tauwehenga hei tatau i te tapeke moni whiwhi, ngā utu rānei mō tētahi wā kua whakaritea, i runga i te mahi a te moni whiwhi, ngā utu rānei mō ia waeine wā.
Ngā Uara Tau: Tikanga Whakatata
Ina he uaua te mahi \( f(x) \) , kāore rānei he taupatupatu tino tika, ka whakamahia ngā tikanga tau hei tatau i te taupū. Ko ngā tikanga noa e whakamahia ana ko:
– Tikanga Riemann: Te whakatau tata i te tauwehenga mā te tāpiri i ngā horahanga o ngā tapawhā hāngai i raro i te kōpiko.
– Tikanga Tāparepare: Te whakatata i te tauwehe mā te tāpiri i ngā horahanga tāparepare i raro i te pihi.
– Te tikanga a Simpson: Ka whakamahia he pūrau tapawhā hei whakatau tata i te horahanga i raro i te kōpiko.
Hei tauira, ko te tikanga trapezoidal mō te tatau i te \( \int_{a}^{b} f(x) \, dx \) me ngā wehenga \( n \) ko:
\[ \int_{a}^{b} f(x) \, dx \approx \frac{ba}{2n} \left[f(x_0) + 2 \sum_{k=1}^{n-1} f(x_k) + f(x_n)\right] \]
ko \( x_0, x_1, …, x_n \) ngā pūwāhi wehewehe o te wā \([a, b]\).
Whakamutunga
Ko te taupū tino he ariā taketake i roto i te tātaitai me ngā tono whānui i roto i ngā mara maha. Mai i te tatau i te horahanga i raro i te pihi ki te rōrahi o ngā totoka hurihuri me te tātari i ngā rahinga ā-tinana me te ōhanga, he taputapu kaha te taupū tino i roto i te whānuitanga o ngā tātaitanga. Mā te whakamahi i ngā tikanga tātari me ngā tikanga tau, ka taea e tātou te aromatawai i ngā taupū tino kia whiwhi ai i ngā hua tika me te whai hua i roto i ngā āhuatanga o te ao tūturu. Mā te māramatanga hohonu ki ngā taupū tino ka huaki te kuaha ki te whakaoti rapanga uaua maha e pā ana ki ngā mahi me ngā horahanga.
