Ajụjụ na Mkparịta ụka Ihe Nlereanya nke Njirimara nke Ihe Ndị A Na-ahụ Anya
Isi ihe dị mkpa bụ echiche dị mkpa na mgbakọ na mwepụ, nke bara uru nke ukwuu n'ọtụtụ ojiji na mgbakọ na mwepụ, fizik, na injinia. N'isiokwu a, anyị ga-akọwa ụfọdụ ihe dị mkpa nke isi ihe dị mkpa ma nye ihe atụ na azịza iji mee ka nghọta gị banyere isiokwu ahụ dịkwuo omimi.
Njirimara nke ihe ndị a kapịrị ọnụ
Tupu anyị abanye n'ihe atụ ndị a, ka anyị lelee ụfọdụ ihe ndị bụ isi nke ihe ndị a kapịrị ọnụ dị mkpa ịmara:
1. Njirimara Linearity:
– Ọ bụrụ na \( f(x) \) na \( g(x) \) bụ ọrụ ndị a na-ejikọta ọnụ na \( a \) na \( b \) bụ ndị na-agbanwe agbanwe, mgbe ahụ:
\[
\int_a^b [af(x) + bg(x)] \, dx = a \int_a^bf(x) \, dx + b \int_a^bg(x) \, dx.
\]
2. Njikọta nke ihe na-agbanweghi agbanwe:
– Ọ bụrụ na \( c \) bụ ihe na-agbanwe agbanwe, mgbe ahụ:
\[
\int_a^bc \, dx = c(b – a).
\]
3. Àgwà nke Mgbakwunye Oge:
\[
\int_a^cf(x) \, dx + \int_c^bf(x) \, dx = \int_a^bf(x) \, dx
\]
4. Mgbanwe nke Oke:
\[
\int_a^bf(x) \, dx = – \int_b^af(x) \, dx
\]
5. Ọ dịghị ihe dị ka otu oke ahụ:
\[
\int_a^af(x) \, dx = 0
\]
Ajụjụ Ihe Nlereanya nke 1: Iji Ihe Njirimara Linearity
Ihe atụ nke nsogbu:
Gbakọọ uru nke:
\[
\int_0^2 (3x^2 + 2x) \, dx
\]
Mkparịta ụka:
Jiri ihe onwunwe linearity kewaa ihe mejupụtara ahụ ụzọ abụọ:
\[
\int_0^2 (3x^2 + 2x) \, dx = \int_0^2 3x^2 \, dx + \int_0^2 2x \, dx
\]
Ka anyị gbakọọ ihe mbụ dị mkpa:
\[
\int_0^2 3x^2 \, dx
\]
\[
= 3 \int_0^2 x^2 \, dx
\]
\[
= 3 \aka ekpe[ \frac{x^3}{3} \nri]_0^2
\]
\[
= 3 \aka ekpe( \frac{2^3}{3} – \frac{0^3}{3} \aka nri)
\]
\[
= 3 \aka ekpe( \frac{8}{3} \aka nri)
\]
\[
= 8
\]
Ugbu a, anyị na-agbakọ ihe nke abụọ:
\[
\int_0^2 2x \, dx
\]
\[
= 2 \int_0^2 x \, dx
\]
\[
= 2 \aka ekpe[ \frac{x^2}{2} \nri]_0^2
\]
\[
= 2 \aka ekpe(1 - 0 \aka nri)
\]
\[
= 2
\]
Jikọta nsonaazụ abụọ ahụ:
\[
\int_0^2 (3x^2 + 2x) \, dx = 8 + 2 = 10
\]
Ajụjụ Ihe Nlereanya nke 2: Integral nke Constant
Ihe atụ nke nsogbu:
Gbakọọ uru nke:
\[
\int_1^4 5 \, dx
\]
Mkparịta ụka:
Site na iji ihe onwunwe dị mkpa nke ihe ndị na-agbanwe agbanwe, anyị nwere ike ide:
\[
\int_1^4 5 \, dx = 5 \cdot (4 – 1)
\]
\[
= 5 \ kdot 3
\]
\[
= 15
\]
Ihe atụ Ajụjụ nke 3: Àgwà nke Mgbanwe Oke
Ihe atụ nke nsogbu:
Gosipụta na:
\[
\int_2^5 x^2 \, dx = – \int_5^2 x^2 \, dx
\]
Mkparịta ụka:
Anyị na-amalite site na njikọta nke \( x^2 \) na oge \( [2, 5] \):
\[
\int_2^5 x^2 \, dx = \ekpe[ \frac{x^3}{3} \nri]_2^5
\]
\[
= \frac{5^3}{3} – \frac{2^3}{3}
\]
\[
= \frac{125}{3} – \frac{8}{3}
\]
\[
= \frac{117}{3}
\]
\[
= 39
\]
Ugbua, ka anyị gbakọọ ihe dị n'ime \(x^2 \) na oge \( [5, 2] \) ma hụ na anyị gbanwere akara nke azịza ahụ:
\[
\int_5^2 x^2 \, dx = \ekpe[ \frac{x^3}{3} \nri]_5^2
\]
\[
= \frac{2^3}{3} – \frac{5^3}{3}
\]
\[
= \frac{8}{3} – \frac{125}{3}
\]
\[
= -\frac{117}{3}
\]
\[
= -39
\]
E gosipụtara na:
\[
\int_2^5 x^2 \, dx = – \int_5^2 x^2 \, dx.
\]
Ihe atụ Ajụjụ nke 4: Àgwà nke Mgbakwunye Oge
Ihe atụ nke nsogbu:
Ọ bụrụ na a maara \(\int_2^4 f(x) \, dx = 7\) na \(\int_4^6 f(x) \, dx = 5\), gbakọọ uru nke \(\int_2^6 f(x) \, dx\).
Mkparịta ụka:
Iji ihe onwunwe mgbakwunye oge:
\[
\int_2^6 f(x) \, dx = \int_2^4 f(x) \, dx + \int_4^6 f(x) \, dx
\]
\[
= 7 + 5
\]
\[
= 12
\]
Mmechi
Ihe dị mkpa nke dị na njikọ ahụ nwere ọtụtụ ihe dị mkpa nke nwere ike inyere anyị aka idozi ụdị nsogbu dị iche iche nke ọma. N'isiokwu a, anyị atụleela ụfọdụ n'ime ihe ndị a bụ isi ma nye ihe atụ ndị na-egosi otu esi etinye ihe ndị a n'ọrụ n'ọrụ. Site na nghọta na omume zuru oke, ị ga-enwe ike idozi nsogbu ndị dị mkpa n'enweghị obi ike.
