Nā nīnau hoʻohālike a me ke kūkākūkā ʻana i nā waiwai o nā Integrals Paʻa
ʻO ka integral paʻa kahi manaʻo nui i ka calculus, he mea pono loa ia i nā ʻano noi like ʻole i ka makemakika, physics, a me ka ʻenekinia. Ma kēia ʻatikala, e wehewehe mākou i kekahi mau waiwai koʻikoʻi o ka integral paʻa a hāʻawi i nā laʻana a me nā hoʻonā e hoʻoikaika i kou ʻike i ke kumuhana.
Nā Waiwai o nā Hoʻohui Paʻa
Ma mua o ko mākou komo ʻana i nā pilikia hoʻohālike, e nānā hou kākou i kekahi mau waiwai kumu o nā integrals definite e pono e ʻike:
1. Waiwai Linearity:
– Inā he mau hana hoʻohui ʻia ʻo \( f(x) \) a me \( g(x) \) a he mau mea mau ʻo \( a \) a me \( b \), a laila:
\[
\int_a^b [af(x) + bg(x)] \, dx = a \int_a^bf(x) \, dx + b \int_a^bg(x) \, dx.
\]
2. Hoʻohui o kahi Kūpaʻa:
– Inā he mea mau ʻo \( c \), a laila:
\[
\int_a^bc \, dx = c(b – a).
\]
3. Nā ʻano o ka hoʻohui ʻana o ka wā:
\[
\int_a^cf(x) \, dx + \int_c^bf(x) \, dx = \int_a^bf(x) \, dx
\]
4. Ka Hoʻohuli ʻana i nā Palena:
\[
\int_a^bf(x) \, dx = – \int_b^af(x) \, dx
\]
5. ʻAʻohe ma ka palena like:
\[
\int_a^af(x) \, dx = 0
\]
Laʻana Nīnau 1: Ke hoʻohana nei i ka Waiwai Linearity
Laʻana o nā pilikia:
E helu i ka waiwai o:
\[
\int_0^2 (3x^2 + 2x) \, dx
\]
Kūkākūkā:
E hoʻohana i ka waiwai linearity e hoʻokaʻawale i ka integral i ʻelua:
\[
\int_0^2 (3x^2 + 2x) \, dx = \int_0^2 3x^2 \, dx + \int_0^2 2x \, dx
\]
E helu kākou i ka integral mua:
\[
\int_0^2 3x^2 \, dx
\]
\[
= 3 \int_0^2 x^2 \, dx
\]
\[
= 3 \left[ \frac{x^3}{3} \right]_0^2
\]
\[
= 3 \left( \frac{2^3}{3} – \frac{0^3}{3} \ʻākau)
\]
\[
= 3 \left( \frac{8}{3} \ʻākau)
\]
\[
= 8
\]
I kēia manawa, ke helu nei mākou i ka integral ʻelua:
\[
\int_0^2 2x \, dx
\]
\[
= 2 \int_0^2 x \, dx
\]
\[
= 2 \left[ \frac{x^2}{2} \right]_0^2
\]
\[
= 2 \hema( 1 – 0 \ʻākau)
\]
\[
= 2
\]
E hoʻohui i nā hopena ʻelua:
\[
\int_0^2 (3x^2 + 2x) \, dx = 8 + 2 = 10
\]
Laʻana Nīnau 2: Hoʻohui o kahi Kūpaʻa
Laʻana o nā pilikia:
E helu i ka waiwai o:
\[
\int_1^4 5 \, dx
\]
Kūkākūkā:
Ma ka hoʻohana ʻana i ka waiwai integral o nā constants, hiki iā mākou ke kākau:
\[
\int_1^4 5 \, dx = 5 \cdot (4 – 1)
\]
\[
= 5 \cdot 3
\]
\[
= 15
\]
Laʻana Nīnau 3: Nā Waiwai o ka Hoʻololi Palena
Laʻana o nā pilikia:
E hōʻoia i kēia:
\[
2^5 x^2 \, dx = – 5^2 x^2 \, dx
\]
Kūkākūkā:
Hoʻomaka mākou me ka integral o \( x^2 \) ma ka wā \( [2, 5] \):
\[
\int_2^5 x^2 \, dx = \left[ \frac{x^3}{3} \right]_2^5
\]
\[
= \frac{5^3}{3} – \frac{2^3}{3}
\]
\[
= \frac{125}{3} – \frac{8}{3}
\]
\[
= \frac{117}{3}
\]
\[
= 39
\]
I kēia manawa, e helu kākou i ka integral o \( x^2 \) ma ka interval \( [5, 2] \) a e hōʻoia e hoʻohuli i ka hōʻailona o ka pane:
\[
\int_5^2 x^2 \, dx = \left[ \frac{x^3}{3} \right]_5^2
\]
\[
= \frac{2^3}{3} – \frac{5^3}{3}
\]
\[
= \frac{8}{3} – \frac{125}{3}
\]
\[
= -\frac{117}{3}
\]
\[
= -39
\]
Ua hōʻoia ʻia:
\[
\int_2^5 x^2 \, dx = – \int_5^2 x^2 \, dx.
\]
Laʻana Nīnau 4: Nā Waiwai o ka Hoʻohui Waena
Laʻana o nā pilikia:
Inā ʻike ʻia ʻo \(\int_2^4 f(x) \, dx = 7\) a me \(\int_4^6 f(x) \, dx = 5\), e helu i ka waiwai o \(\int_2^6 f(x) \, dx\).
Kūkākūkā:
Ke hoʻohana nei i ka waiwai hoʻohui interval:
\[
2^6 f(x) \, dx = 2^4 f(x) \, dx + 4^6 f(x) \, dx
\]
\[
= 7 + 5
\]
\[
= 12
\]
Ka hopena
He nui nā ʻano koʻikoʻi o ka integral paʻa e hiki ke kōkua iā mākou e hoʻoponopono i nā ʻano pilikia like ʻole me ka ʻoi aku ka maikaʻi. Ma kēia ʻatikala, ua kūkākūkā mākou i kekahi o kēia mau ʻano kumu a hāʻawi i nā laʻana e hōʻike ana pehea e hiki ai ke hoʻopili ʻia kēia mau ʻano i ka hana. Me ka lawa ʻana o ka ʻike a me ka hoʻomaʻamaʻa ʻana, hiki iā ʻoe ke hoʻoponopono i nā pilikia integral paʻa me ka hilinaʻi nui aʻe.
