Ngā Tauira Pātai me te Kōrero mō ngā Āhuatanga o ngā Taurite Tūturu
He ariā taketake te taupū tino i roto i te tātaitai, he tino whai hua i roto i te whānuitanga o ngā tono i roto i te pāngarau, te ahupūngao, me te hangarau. I roto i tēnei tuhinga, ka whakamāramahia e mātou ētahi āhuatanga nui o te taupū tino, me te whakarato tauira me ngā otinga hei whakahōhonu ake i tō māramatanga ki te kaupapa.
Ngā Āhuatanga o ngā Taurite Tūturu
I mua i te urunga atu ki ngā tauira rapanga, me arotake tātou i ētahi āhuatanga taketake o ngā tauwehenga tino tika hei mōhio:
1. Āhuatanga Raina:
– Mena he mahi whakauru a \( f(x) \) me \( g(x) \) ā, he pūmau a \( a \) me \( b \), kāti:
\[
\int_a^b [af(x) + bg(x)] \, dx = a \int_a^bf(x) \, dx + b \int_a^bg(x) \, dx.
\]
2. Te Whakapūmau o tētahi Pūmau:
– Mena he pūmau te \( c \), kāti:
\[
\int_a^bc \, dx = c(b – a).
\]
3. Ngā Āhuatanga o te Tāpiritanga Āputa:
\[
\int_a^cf(x) \, dx + \int_c^bf(x) \, dx = \int_a^bf(x) \, dx
\]
4. Te Hurihanga o ngā Here:
\[
\int_a^bf(x) \, dx = – \int_b^af(x) \, dx
\]
5. Kore i te rohenga kotahi:
\[
\int_a^af(x) \, dx = 0
\]
Tauira Pātai 1: Te Whakamahi i te Āhuatanga Raina
Tauira raruraru:
Tātaihia te uara o:
\[
\int_0^2 (3x^2 + 2x) \, dx
\]
Kōrero:
Whakamahia te āhuatanga rārangi hei wehe i te tauwehenga kia rua:
\[
\int_0^2 (3x^2 + 2x) \, dx = \int_0^2 3x^2 \, dx + \int_0^2 2x \, dx
\]
Me tatau te taupū tuatahi:
\[
\int_0^2 3x^2 \, dx
\]
\[
= 3 \int_0^2 x^2 \, dx
\]
\[
= 3 \left[ \frac{x^3}{3} \right]_0^2
\]
\[
= 3 \left( \frac{2^3}{3} – \frac{0^3}{3} \right)
\]
\[
= 3 \left( \frac{8}{3} \right)
\]
\[
= 8
\]
Nā, ka tatauhia e tātou te taupū tuarua:
\[
\int_0^2 2x \, dx
\]
\[
= 2 \int_0^2 x \, dx
\]
\[
= 2 \left[ \frac{x^2}{2} \right]_0^2
\]
\[
= 2 \maui( 1 – 0 \matau)
\]
\[
= 2
\]
Whakakotahitia ngā hua e rua:
\[
\int_0^2 (3x^2 + 2x) \, dx = 8 + 2 = 10
\]
Tauira Pātai 2: Taupū o tētahi Pūmau
Tauira raruraru:
Tātaihia te uara o:
\[
\int_1^4 5 \, dx
\]
Kōrero:
Mā te whakamahi i te āhuatanga whakauru o ngā pūmau, ka taea e tātou te tuhi:
\[
\int_1^4 5 \, dx = 5 \cdot (4 – 1)
\]
\[
= 5 \cdot 3
\]
\[
= 15
\]
Tauira Pātai 3: Ngā Āhuatanga o te Huringa Here
Tauira raruraru:
Whakamātauria:
\[
\int_2^5 x^2 \, dx = – \int_5^2 x^2 \, dx
\]
Kōrero:
Ka tīmata tātou me te taupū o \( x^2 \) i te wā \( [2, 5] \):
\[
\int_2^5 x^2 \, dx = \left[ \frac{x^3}{3} \right]_2^5
\]
\[
= \frac{5^3}{3} – \frac{2^3}{3}
\]
\[
= \frac{125}{3} – \frac{8}{3}
\]
\[
= \frac{117}{3}
\]
\[
= 39
\]
Nā, me tatau te taupū o \( x^2 \) i runga i te wā \( [5, 2] \) ā, kia tino hurihia te tohu o te whakautu:
\[
\int_5^2 x^2 \, dx = \left[ \frac{x^3}{3} \right]_5^2
\]
\[
= \frac{2^3}{3} – \frac{5^3}{3}
\]
\[
= \frac{8}{3} – \frac{125}{3}
\]
\[
= -\frac{117}{3}
\]
\[
= -39
\]
Kua whakamātauhia:
\[
\int_2^5 x^2 \, dx = – \int_5^2 x^2 \, dx.
\]
Tauira Pātai 4: Ngā Āhuatanga o te Tāpiritanga Āputa
Tauira raruraru:
Mena e mōhiotia ana te uara o \(\int_2^4 f(x) \, dx = 7\) me te \(\int_4^6 f(x) \, dx = 5\), tatauhia te uara o \(\int_2^6 f(x) \, dx\).
Kōrero:
Mā te whakamahi i te āhuatanga tāpiri āputa:
\[
\int_2^6 f(x) \, dx = \int_2^4 f(x) \, dx + \int_4^6 f(x) \, dx
\]
\[
= 7 + 5
\]
\[
= 12
\]
Whakamutunga
He maha ngā āhuatanga nui o te taupū tino hei āwhina i a tātou ki te whakaoti rapanga maha me te whai hua ake. I roto i tēnei tuhinga, kua matapakihia e mātou ētahi o ēnei āhuatanga taketake, ā, kua whakaratohia he tauira e whakaatu ana me pēhea te whakamahi i ēnei āhuatanga i roto i te mahi. Mēnā he māramatanga me te mahi, ka taea e koe te whakaoti rapanga taupū tino me te māia ake.
