Nā Hoʻohui Paʻa a me nā Hoʻohui Paʻa ʻole
ʻO ka hoʻohui ʻana he manaʻo nui ia i ka calculus, i hoʻohana ʻia e helu i ka ʻāpana ma lalo o kahi piʻo, ka huina cumulative, a me kekahi mau noi ʻē aʻe ma nā kahua e like me ke kino, ka hoʻokele waiwai, ka biology, a me ka ʻenekinia. Manaʻo pinepine ʻia ka hoʻohui ʻana ʻo ia ka hana inverse o ka differentiation. Ma kēia ʻano, inā ʻike mākou i ka derivative o kahi hana, hiki iā mākou ke loaʻa ka hana mua me ka hoʻohana ʻana i ka integral. Aia ʻelua mau ʻano integrals kaulana a hoʻohana pinepine ʻia i ka calculus: ka integral definite a me ka integral indefinite. E ʻeli hohonu kēia ʻatikala i nā ʻano integrals ʻelua, e hāʻawi ana i nā wehewehe kūhelu, nā laʻana, a me nā noi kūpono.
Hoʻohui palena ʻole
Wehewehewehe
ʻO ka integral indefinite kahi hana e hoʻihoʻi ana i ka hana mua mai kahi derivative. Ua hōʻike ʻia ka integral indefinite o ka hana \( f(x) \) penei:
\[ \int f(x) \, dx = F(x) + C \]
Ma hea:
– ʻO \( ʻO F(x) \) kahi hana mua (antiderivative) o \( f(x) \).
– ʻO \( C \) ke kūpaʻa o ka hoʻohui ʻana.
Ke kamaʻilio mākou e pili ana i nā integrals indefinite, ke ʻimi nei mākou i kahi ʻohana o nā hana nona ka derivative mua ʻo \( f(x) \).
Eia kekahi laʻana
Manaʻo mākou makemake mākou e ʻimi i ka integral indefinite o ka hana \( f(x) = 2x \). Ke ʻimi nei mākou i kahi hana \( F(x) \) i hiki ai i ka derivative o \( F(x) \) ʻo \( 2x \). Ua ʻike mākou ʻo ka derivative o \( x^2 \) ʻo \( 2x \), no laila ʻo kekahi o nā hana mua o \( 2x \) ʻo \( x^2 \). Eia naʻe, pono mākou e hoʻohui i kahi constant o ka integration \( C \), no ka mea, ʻo ka derivative o kahi constant he zero. No laila,
\[ \int 2x \, dx = x^2 + C \]
ʻO kekahi laʻana: ʻo ka integral o \( e^x \) ʻo \( e^x + C \), no ka mea, ʻo ka derivative o \( e^x \) e mau ana ʻo \( e^x \).
Hoʻohui Paʻa
Wehewehewehe
He mau palena hoʻohui ko kahi integral paʻa a he helu maoli kona hopena. Ua hōʻike ʻia ka integral paʻa o \( f(x) \) mai \( a \) a i \( b \) penei:
\[ \int_{a}^{b} f(x) \, dx \]
Hoʻomaulia kēia hoʻohui i ka “ʻāpana” ma lalo o ke kūlou \( f(x) \) mai \( x = a \) a i \( x = b \).
Ke Kumumanaʻo Kumu o ka Calculus
Hoʻopili ka Theorem Kumu o Calculus i nā integrals paʻa a me nā integrals pau ʻole. Ke ʻōlelo nei kēia theorem inā he hana mua ʻo \( F \) o \( f \), a laila:
\[ \int_{a}^{b} f(x) \, dx = F(b) – F(a) \]
Eia kekahi laʻana
Manaʻo mākou e makemake mākou e ʻimi i ka integral paʻa o \( f(x) = 2x \) ma ka wā [1, 3]. ʻO ka mea mua, pono mākou e ʻimi i ka hana mua o \( 2x \), ʻo ia hoʻi ʻo \( x^2 \). Ma o ke Kumumanaʻo Kumu o ka Calculus, hiki iā mākou ke helu:
\[ \int_{1}^{3} 2x \, dx = \left[ x^2 \right]_{1}^{3} = 3^2 – 1^2 = 9 – 1 = 8 \]
No laila, ʻo ka integral paʻa \( \int_{1}^{3} 2x \, dx \) he 8.
Nā Hoʻohana Kūpono
He nui nā noi o nā Integrals ma nā ʻano like ʻole. ʻO kekahi o kēia mau mea:
ʻO ke kinoea
I ke ʻano physics, hoʻohana ʻia nā integrals e helu i nā mea like ʻole, e like me:
- Ka neʻe ʻana o nā mea ma nā wikiwiki like ʻole.
– Uila a me ka magnetism, ʻoi aku hoʻi i ka helu ʻana i nā kahua uila a i ʻole nā kahua magnetic.
– Hana i hana ʻia e kahi ikaika loli.
Eia kekahi laʻana, inā hāʻawi ʻia ka ikaika \( F(x) \) ma ke ʻano he hana o ka mamao \( x \), a laila hiki ke helu ʻia ka hana i hana ʻia e kēia ikaika mai ke kūlana \( a \) a i \( b \) me ka hoʻohana ʻana i kahi integral paʻa:
\[ W = \int_{a}^{b} F(x) \, dx \]
hoʻokele waiwai
I loko o ka hoʻokele waiwai, hoʻohana ʻia nā integrals e helu i nā kumukūʻai holoʻokoʻa, nā loaʻa kālā, a i ʻole nā loaʻa ma lalo o nā kūlana loli. No ka laʻana, inā hāʻawi ʻia ke kumukūʻai marginal \( C'(x) \) ma ke ʻano he hana o ka nui o nā waiwai i hana ʻia, a laila hiki ke loaʻa ke kumukūʻai holoʻokoʻa \( C(x) \) ma ka hoʻohui ʻana i ka hana kumukūʻai marginal.
ʻO ke olaola
I loko o ka biology, hiki ke hoʻohana ʻia nā integrals e hoʻohālike i ka ulu ʻana o ka heluna kanaka. E ʻo \( r(t) \) ka helu ulu ʻana o ka heluna kanaka ma ke ʻano he hana o ka manawa \( t \), a laila ʻo ka heluna kanaka holoʻokoʻa i ka manawa \( t \) ka integral o \( r(t) \) mai ka manawa mua a hiki i \( t \).
Nā Hana Helu
I nā hihia hana he nui, ʻoiai inā paʻakikī a hiki ʻole paha ke hoʻohui ʻia ka hana i hoʻohui ʻia ma ke ʻano analytical, hoʻohana ʻia nā ʻano helu e like me ke ʻano trapezoidal a i ʻole ke ʻano Simpson e hoʻokokoke i ka integral. ʻO kekahi o ia ʻano hoʻokokoke helu ʻo ia ke ʻano lula trapezoidal.
ʻAno Trapezoidal
Hoʻokokoke ke ʻano trapezoidal i ka ʻāpana ma lalo o kahi piʻo ma ka hōʻuluʻulu ʻana i nā ʻāpana o nā trapezoids i hana ʻia ma waena o nā kiko i wehewehe ʻia ma ka hana. Ma ke ʻano makemakika:
\[ \int_{a}^{b} f(x) \, dx \approx \frac{ba}{2} \left[ f(a) + f(b) \ʻākau] \]
No ka pololei ʻoi aku, hiki ke puʻunaue ʻia ka wā [a, b] i kekahi mau wā liʻiliʻi.
Ke ʻano hana a Simpson
Hoʻokokoke pū ke ʻano hana a Simpson i ka integral ma ka puʻunaue ʻana i ka interval i nā subintervals a me ka hoʻohana ʻana i ka polynomial kekelē lua e hoʻokokoke i ka hana \( f(x) \). Hāʻawi kēia i nā hopena pololei ma mua o ke ʻano trapezoidal.
Hoʻohui a me ka Hoʻololi
Ma nā kahua makemakika e like me ka loiloi Fourier lāua ʻo Laplace, he mau mea hana koʻikoʻi nā integrals. Hoʻohana ʻia kēia mau hoʻololi e hoʻololi i nā hana mai ka domain manawa a i ka domain alapine, e hāʻawi ana i nā ʻike hohonu i ka loiloi ʻōnaehana linear, ka loiloi hōʻailona, a me ka hana kiʻi.
Ka hopena
He mau mea hana koʻikoʻi nā hoʻohui i ka calculus me nā noi ākea ma nā ʻano aʻo like ʻole. Hāʻawi nā hoʻohui pau ʻole i nā hopena laulā e pili ana i nā hana mua a me kahi paʻa o ka hoʻohui ʻana, ʻoiai hoʻohana ʻia nā hoʻohui paʻa e helu i nā waiwai helu paʻa ma waena o nā palena i hāʻawi ʻia. ʻO ka hoʻomaopopo ʻana i kēia mau ʻano hoʻohui ʻelua e wehe i ka puka i nā noi he nui, mai ka hoʻoponopono ʻana i nā pilikia physics a hiki i ka hoʻonui ʻana i nā hiʻohiʻona hoʻokele waiwai. ʻO nā ʻano helu, e like me ke ʻano trapezoidal a me ke ʻano o Simpson, he kuleana koʻikoʻi ke hiki ʻole ka hoʻohui analytical. ʻO ke kumu, hāʻawi nā hoʻohui i kahi ʻōnaehana ikaika no ka hoʻomaopopo ʻana a me ke kumu hoʻohālike i nā hanana kūlohelohe a me nā mea i hana ʻia e ke kanaka.
