Nā nīnau hoʻohālike a me ke kūkākūkā ʻana o ka hoʻohana ʻana o nā Integrals i ka helu ʻana i ka ʻāpana o kahi mokulele pālahalaha
I ke aʻo ʻana i ka makemakika, ʻike pinepine ʻia nā integrals i ka calculus. ʻO kekahi o nā hoʻohana kaulana loa o nā integrals ʻo ia ka helu ʻana i ka ʻāpana ma lalo o kahi piʻo a i ʻole ka mokulele. E kūkākūkā kēia ʻatikala i kekahi mau pilikia hoʻohālike a kūkākūkā i ka hoʻohana ʻana o nā integrals i ka helu ʻana i ka ʻāpana o kahi mokulele.
Hoʻolauna i ke Kumumanaʻo
Ma mua o ka neʻe ʻana i ka pilikia hoʻohālike, e nānā hou kākou i ke kumumanaʻo kumu o ka helu ʻana i ka ʻāpana ma lalo o kahi piʻo me ka hoʻohana ʻana i nā integrals. Inā loaʻa iā mākou kahi hana f(x) e hoʻomau ana ma ka manawa [a, b], a laila ʻo ka ʻāpana ma lalo o ke piʻo y = f(x) mai x = a a i x = b penei:
\[ L = \int_{a}^{b} f(x) \, dx \]
Ma ke ʻano geometric, ʻo ia hoʻi ke hōʻuluʻulu nei mākou i ka ʻāpana o kahi huinahā lahilahi loa mai x = a a i x = b.
Laʻana Nīnau 1
Nīnau
E helu i ka ʻāpana ma lalo o ke kūlou y = x² ma ka wā [1, 3].
Pahana
No ka helu ʻana i ka wahi, hoʻohana mākou i ka integral:
\[ L = \int_{1}^{3} x^2 \, dx \]
Hoʻomaka mākou ma ka ʻimi ʻana i ka antiderivative o \( x^2 \). ʻO ka antiderivative o \( x^2 \) ʻo \( \frac{x^3}{3} \). A laila lilo ka integral:
\[ L = \left[ \frac{x^3}{3} \right]_{1}^{3} \]
E hoʻomanaʻo he pono kākou e loiloi i ka antiderivative ma nā palena o ka integral:
\[ L = \left( \frac{3^3}{3} \ʻākau) – \left( \frac{1^3}{3} \ʻākau) \]
\[ L = \left( \frac{27}{3} \ʻākau) – \left( \frac{1}{3} \ʻākau) \]
\[ L = 9 – \frac{1}{3} \]
\[ L = \frac{27}{3} – \frac{1}{3} \]
\[ L = \frac{26}{3} \]
No laila, ʻo ka ʻāpana ma lalo o ke piʻo y = x² mai x = 1 a i x = 3 penei:
\[ \frac{26}{3} \, \text{ʻāpana ʻāpana} \]
Laʻana Nīnau 2
Nīnau
E hoʻoholo i ka ʻāpana o ka ʻāpana i kaupalena ʻia e ka piʻo y = x³ a me nā laina x = 1 a me x = 2.
Pahana
No ka helu ʻana i ka wahi, hoʻohana mākou i ka integral:
\[ L = \int_{1}^{2} x^3 \, dx \]
E like me ka mea maʻamau, hoʻomaka mākou ma ka ʻimi ʻana i ka antiderivative o \( x^3 \). ʻO ka antiderivative o \( x^3 \) ʻo \( \frac{x^4}{4} \). ʻO ka integral e lilo i:
\[ L = \left[ \frac{x^4}{4} \right]_{1}^{2} \]
E loiloi i nā palena o ka integral:
\[ L = \left( \frac{2^4}{4} \ʻākau) – \left( \frac{1^4}{4} \ʻākau) \]
\[ L = \left( \frac{16}{4} \ʻākau) – \left( \frac{1}{4} \ʻākau) \]
\[ L = 4 – \frac{1}{4} \]
\[ L = \frac{16}{4} – \frac{1}{4} \]
\[ L = \frac{15}{4} \]
No laila, ʻo ka ʻāpana ma lalo o ke piʻo y = x³ mai x = 1 a i x = 2 penei:
\[ \frac{15}{4} \, \text{ʻāpana ʻāpana} \]
Laʻana Nīnau 3
Nīnau
E hoʻoholo i ka ʻāpana o ka ʻāpana i kaupalena ʻia e nā piʻo y = x² + 1 a me y = 2x + 2 ma ka wā x = 0 a i x = 1.
Pahana
ʻO ka mea mua, pono mākou e ʻimi i nā kiko intersection e hoʻoholo ai i nā palena o ka hoʻohui ʻana. ʻO ka hopena i \( x^2 + 1 = 2x + 2 \):
\[ x^2 + 1 = 2x + 2 \]
\[ x^2 – 2x – 1 = 0 \]
Ke hoʻohana nei i ke ʻano quadratic:
\[ x = \frac{2 \pm \sqrt{4 + 4}}{2} \]
\[ x = \frac{2 \pm \sqrt{8}}{2} \]
\[ x = \frac{2 \pm 2\sqrt{2}}{2} \]
\[ x = 1 \pm \sqrt{2} \]
Eia nō naʻe, no nā palena kiʻekiʻe a me lalo ma waena o 0 a me 1, ʻaʻole pono mākou e hoʻohana i ka hopena quadratic, ʻo ka palena integral maʻamau mai 0 a 1. A laila, e helu i ka ʻāpana o ka piʻo y luna me ka hoʻemi ʻana i ka piʻo y lalo e like me kēia mau palena:
\[ L = \int_{0}^{1} [(2x + 2) – (x^2 + 1)] \, dx \]
Hoʻomaʻalahi hana:
\[ L = \int_{0}^{1} (2x + 2 – x^2 – 1) \, dx \]
\[ L = \int_{0}^{1} (-x^2 + 2x + 1) \, dx \]
A laila, ʻike mākou i ka antiderivative:
ʻO ka antiderivative o \( (-x^2) \) ʻo \( -\frac{x^3}{3} \),
ʻO ka antiderivative o \( (2x) \) ʻo \( x^2 \),
ʻO ka antiderivative o \( (1) \) ʻo \( x \).
No laila,
\[ L = \left. \left(-\frac{x^3}{3} + x^2 + x \right) \right|_0^1 \]
Ka loiloi aʻe:
\[ L = \left[ -\frac{1^3}{3} + 1^2 + 1 \ʻākau] – \left[ -\frac{0^3}{3} + 0^2 + 0 \ʻākau] \]
\[ L = \left[ -\frac{1}{3} + 1 + 1 \ʻākau] – \left[ 0 \ʻākau] \]
\[ L = -\frac{1}{3} + 2 \]
\[ L = \frac{6}{3} – \frac{1}{3} \]
\[ L = \frac{5}{3} \]
No laila, ʻo ka ʻāpana o ka ʻāina i kaupalena ʻia e nā piʻo y = x² + 1 a me y = 2x + 2 ma ka wā [0, 1] penei:
\[ \frac{5}{3} \, \text{ʻāpana ʻāpana} \]
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Mai nā hiʻohiʻona ma luna, hiki iā kākou ke ʻike pehea e hiki ai ke hoʻohana ʻia nā integrals e helu i ka ʻāpana ma lalo o kahi piʻo a i ʻole ma waena o ʻelua mau piʻo. Me ka hoʻomaopopo pono ʻana i nā manaʻo kumu o nā integrals a me nā ʻenehana antiderivative, lilo ka helu ʻana i kēia mau wahi i ʻōnaehana a me ka pono. Manaʻolana, ua hoʻonui kēia ʻatikala i ko kākou ʻike i ka hoʻohana ʻana o nā integrals i ke ao maoli, ʻoi aku hoʻi ma ke kahua o ke ana ʻana i ka ʻāpana o nā ʻili mokulele.
