Nā Laʻana o nā Nīnau a me ke Kūkākūkā ʻana o nā Hoʻohui Paʻa
ʻO ka integral paʻa kahi manaʻo koʻikoʻi i ka calculus, i hoʻohana pinepine ʻia e ʻimi i ka wahi ma lalo o kahi piʻo, e helu i ka nui o nā mea paʻakikī, a no nā noi ʻē aʻe he nui i ka ʻenekinia a me ka physics. ʻO ke kūkākūkā ʻana i ka integral paʻa ʻaʻole wale e hāʻawi i kahi ʻike kumu o kēia manaʻo akā e hoʻoikaika pū i kā mākou mau mākau loiloi makemakika. Manaʻo kēia ʻatikala e hāʻawi i nā laʻana o nā pilikia integral paʻa me nā kūkākūkā kikoʻī.
Manaʻo Kumu o ka Hoʻohui Paʻa
Ma mua o ko kākou komo ʻana i nā pilikia hoʻohālike, e nānā hou kākou i kekahi mau manaʻo kumu o nā integrals paʻa. ʻO ka integral paʻa, i hōʻike ʻia e \(\int_a^bf(x) \, dx\), e hōʻike ana i ka ʻāpana ma lalo o ke kūlou o ka hana \(f(x)\) mai ke kiko \(x = a\) a i ke kiko \(x = b\).
Ma ke ʻano makemakika, hiki ke hōʻike ʻia ka integral paʻa mai \(a\) a i \(b\) o ka hana \(f(x)\) penei:
\[ \int_a^bf(x) \, dx = F(b) – F(a) \]
kahi ʻo \(F(x)\) ka antiderivative o \(f(x)\).
Nā Nīnau Laʻana a me ke Kūkākūkā
E nānā kākou i kekahi mau laʻana o nā pilikia hoʻohui paʻa a me kā lākou mau kūkākūkā ʻana.
Laʻana Nīnau 1
Nīnau:
E helu i ka hoʻohui paʻa o ka hana \(f(x) = 2x\) mai \(x = 1\) a i \(x = 3\).
Kūkākūkā:
No ka hoʻoponopono ʻana i kēia integral, ʻike mua mākou i ka antiderivative o \(f(x) = 2x\).
ʻO ke antiderivative o \(2x\) penei:
\[ F(x) = x^2 + C \]
Eia nō naʻe, i loko o nā integrals paʻa, ʻaʻole pono mākou i ke kūpaʻa o ka hoʻohui ʻana \(C\).
I kēia manawa, e hoʻohana i nā palena o nā integrals e helu ai:
\[ \int_1^3 2x \, dx = F(3) – F(1) \]
E helu i ka waiwai o \(F(x)\) ma kēia mau palena:
\[ F(3) = 3^2 = 9 \]
\[ F(1) = 1^2 = 1 \]
No laila,
\[ \int_1^3 2x \, dx = 9 – 1 = 8 \]
Laʻana Nīnau 2
Nīnau:
E helu i ka hoʻohui paʻa o ka hana \(f(x) = x^2 + 1\) mai \(x = 0\) a i \(x = 2\).
Kūkākūkā:
E huli i ka antiderivative o \(f(x) = x^2 + 1\).
ʻO ke antiderivative o \(x^2\) penei:
\[ \frac{1}{3}x^3 \]
ʻO ka antiderivative o \(1\) ʻo \(x\).
No laila, ʻo ka antiderivative o \(f(x)\) penei:
\[ F(x) = \frac{1}{3}x^3 + x \]
I kēia manawa, e hoʻohana i nā palena o nā integrals e helu ai:
\[ \int_0^2 (x^2 + 1) \, dx = F(2) – F(0) \]
E helu i ka waiwai o \(F(x)\) ma kēia mau palena:
\[ F(2) = \frac{1}{3}(2)^3 + 2 = \frac{8}{3} + 2 = \frac{8}{3} + \frac{6}{3} = \frac{14}{3} \]
\[ F(0) = \frac{1}{3}(0)^3 + 0 = 0 \]
No laila,
\[ \int_0^2 (x^2 + 1) \, dx = \frac{14}{3} – 0 = \frac{14}{3} \]
Laʻana Nīnau 3
Nīnau:
E helu i ka hoʻohui paʻa o ka hana \(f(x) = e^x\) mai \(x = 1\) a i \(x = 2\).
Kūkākūkā:
E huli i ka antiderivative o \(f(x) = e^x\).
ʻO ka antiderivative o \(e^x\) ʻo \(e^x\).
I kēia manawa, e hoʻohana i nā palena o nā integrals e helu ai:
\[ \int_1^2 e^x \, dx = F(2) – F(1) \]
E helu i ka waiwai o \(F(x)\) ma kēia mau palena:
\[ F(2) = e^2 \]
\[ F(1) = e^1 = e \]
No laila,
\[ \int_1^2 e^x \, dx = e^2 – e \]
Laʻana Nīnau 4
Nīnau:
E helu i ka hoʻohui paʻa o ka hana \(f(x) = \sin(x)\) mai \(x = 0\) a i \(x = \pi\).
Kūkākūkā:
E huli i ka antiderivative o \(f(x) = \sin(x)\).
ʻO ke antiderivative o \(\sin(x)\) ʻo \(-\cos(x)\).
I kēia manawa, e hoʻohana i nā palena o nā integrals e helu ai:
\[ \int_0^\pi \sin(x) \, dx = F(\pi) – F(0) \]
E helu i ka waiwai o \(F(x)\) ma kēia mau palena:
\[ F(\pi) = -\cos(\pi) = -(-1) = 1 \]
\[ F(0) = -\cos(0) = -1 \]
No laila,
\[ \int_0^\pi \sin(x) \, dx = 1 – (-1) = 1 + 1 = 2 \]
Laʻana Nīnau 5
Nīnau:
E helu i ka hoʻohui paʻa o ka hana \(f(x) = \frac{1}{x}\) mai \(x = 1\) a i \(x = e\).
Kūkākūkā:
E huli i ka antiderivative o \(f(x) = \frac{1}{x}\).
ʻO ka antiderivative o \(\frac{1}{x}\) ʻo \(\ln|x|\).
I kēia manawa, e hoʻohana i nā palena o nā integrals e helu ai:
\[ \int_1^e \frac{1}{x} \, dx = F(e) – F(1) \]
E helu i ka waiwai o \(F(x)\) ma kēia mau palena:
\[ F(e) = \ln(e) = 1 \]
\[ F(1) = \ln(1) = 0 \]
No laila,
\[ \int_1^e \frac{1}{x} \, dx = 1 – 0 = 1 \]
Ka hopena
Ma o nā hiʻohiʻona ma luna, ua hoʻomaʻamaʻa mākou i ka loaʻa ʻana o nā integrals paʻa o nā hana kumu like ʻole. I kēlā me kēia ʻanuʻu, he mea nui e loaʻa mua ka antiderivative a laila e hoʻohana i nā palena o ka integral e loaʻa ai ka waiwai hope loa.
He kuleana koʻikoʻi ko nā hoʻohui paʻa i nā ʻano haʻawina he nui a me nā noi hana. ʻO ka hoʻomaopopo ʻana i kēia manaʻo a me ka hoʻomaʻamaʻa ʻana me nā laʻana like ʻole e hoʻoikaika nui i kāu mau mākau makemakika.
