Aniq integral va yuza

⭐ — imtihonda eng ko'p ishlatiladigan formulalar. · 🟢 — masalani tezlashtiradi · 🟡 — ilg'or daraja · 🔵 — chuqur o'rganish (olimpiada darajasi)

Aniq integral va yuza

• Egri chiziq ostidagi yuza $A$ uchun $\dfrac{dA}{dx} = y$. Shuning uchun yuza integrallash orqali topiladi.
• Yuza — ingichka to'g'ri to'rtburchaklar yig'indisining limiti: $A = \lim_{\delta x \to 0} \sum y\,\delta x = \int_a^b y\,dx$
• ⭐ Aniq integral: $\displaystyle\int_a^b f(x)\,dx = \big[F(x)\big]_a^b = F(b) - F(a)$, bu yerda $F'(x) = f(x)$.
• Chegaralarni almashtirsangiz, ishorasi o'zgaradi: $\displaystyle\int_b^a f(x)\,dx = -\int_a^b f(x)\,dx$
• ⭐ Egri chiziq va $x$ o'qi orasidagi yuza: $\displaystyle A = \int_a^b y\,dx$
• ⭐ $x$ o'qidan pastdagi yuza uchun integral manfiy chiqadi. Soha o'qning ikki tomonida bo'lsa, integralni ildizlar bo'yicha bo'lib hisoblang: $A = \left|\int_a^p f(x)\,dx\right| + \left|\int_p^b f(x)\,dx\right| \qquad (f(p) = 0)$
• ⭐ Ikki egri chiziq orasidagi yuza (yuqoridagidan pastdagini ayirasiz): $A = \int_a^b \big(f(x) - g(x)\big)\,dx$ Chegaralar $a$, $b$ — egri chiziqlar kesishgan nuqtalar: $f(x) = g(x)$ ni yechasiz.
• Egri chiziq va $y$ o'qi orasidagi yuza: $\displaystyle A = \int_p^q x\,dy$

Xosmas integrallar (improper)

• Chegara cheksiz bo'lsa, limit orqali hisoblanadi: $\int_a^\infty f(x)\,dx = \lim_{b \to \infty} \int_a^b f(x)\,dx$
• Funksiya chegarada aniqlanmagan bo'lsa ham limit olinadi: $\int_0^1 \frac{1}{\sqrt{x}}\,dx = \lim_{a \to 0}\int_a^1 x^{-\frac{1}{2}}\,dx = \lim_{a \to 0}\big(2 - 2\sqrt{a}\big) = 2$

🟢 Asosiy xossalar

$\int_a^b f(x)\,dx + \int_b^c f(x)\,dx = \int_a^c f(x)\,dx \qquad \int_a^a f(x)\,dx = 0$

🟢 Simmetrik chegaralar

$\int_{-a}^{a} f(x)\,dx = 0 \;\;(f \text{ toq}) \qquad \int_{-a}^{a} f(x)\,dx = 2\int_0^a f(x)\,dx \;\;(f \text{ juft})$

Misol: $\displaystyle\int_{-2}^{2} (x^3 + 5x)\,dx = 0$ — hisoblamasdan.

🟡 Funksiyaning o'rtacha qiymati

$\bar{f} = \frac{1}{b - a}\int_a^b f(x)\,dx$

🔵 Integral — yig'indining limiti

$\int_a^b f(x)\,dx = \lim_{\delta x \to 0}\sum f(x_i)\,\delta x$

🟢 Parabolik segment yuzi

$y = a(x - \alpha)(x - \beta)$ va $x$ o'qi orasidagi yuza:

$S = \frac{|a|\,(\beta - \alpha)^3}{6}$

Shu formula parabola $y = ax^2 + \dots$ va istalgan to'g'ri chiziq orasidagi yuza uchun ham ishlaydi: $\alpha$, $\beta$ — kesishish nuqtalarining $x$ koordinatalari. Misol: $y = x^2$ va $y = 4$ orasida: $\alpha = -2$, $\beta = 2$, $S = \dfrac{1 \cdot 4^3}{6} = \dfrac{32}{3}$ ✓

🟢 Birlik kvadratdagi yuzalar

$\int_0^1 x^n\,dx = \frac{1}{n + 1}$

Masalan, $y = x^2$ birlik kvadratni $\frac{1}{3}$ va $\frac{2}{3}$ ga bo'ladi.

🟢 Darajali funksiyalar uchun

$\int_1^\infty \frac{dx}{x^n} = \frac{1}{n - 1} \;\;(n > 1) \qquad \int_0^1 \frac{dx}{x^n} = \frac{1}{1 - n} \;\;(n < 1)$

$n \le 1$ bo'lsa birinchisi, $n \ge 1$ bo'lsa ikkinchisi uzoqlashadi (cheksiz).

🔵 Egri chiziq uzunligi

$L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2}\,dx$