Aylanish jismlarining hajmi

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

Aylanish jismining hajmi

• ⭐ $x$ o'qi atrofida: $\displaystyle V = \int_a^b \pi y^2\,dx$ (jismni qalinligi $\delta x$ bo'lgan disklarga bo'lamiz: $\delta V = \pi y^2\,\delta x$)
• ⭐ $y$ o'qi atrofida: $\displaystyle V = \int_p^q \pi x^2\,dy$

Misol: $y = x$, $0 \le x \le 1$, $x$ o'qi atrofida: $V = \pi\displaystyle\int_0^1 x^2\,dx = \frac{\pi}{3}$ (konus hajmi).

Ikki egri chiziq orasidagi soha aylantirilganda hajm

$V = \pi \int_a^b \big(y_1^2 - y_2^2\big)\,dx \qquad (y_1 \text{ — tashqi}, \; y_2 \text{ — ichki})$

⚠️ $\pi\displaystyle\int (y_1 - y_2)^2\,dx$ noto'g'ri! Avval kvadratlab, keyin ayiring.

🟢 Boshqa o'q atrofida aylantirish

$y = k \text{ atrofida: } V = \pi\int_a^b (y - k)^2\,dx \qquad x = h \text{ atrofida: } V = \pi\int_p^q (x - h)^2\,dy$

🟢 Ikki egri chiziq orasidagi soha (halqa usuli)

$V = \pi\int_a^b \left(y_{\text{tashqi}}^2 - y_{\text{ichki}}^2\right)dx$

🔵 Silindrik qobiq usuli ($y$ o'qi atrofida)

$V = 2\pi\int_a^b x\,y\,dx$

🟢 Integral orqali keltirib chiqariladigan hajmlar