6.4-bo'lim: Silindrli hajm

Quyidagi masalalarning har biri uchun silindrlar usulidan foydalanib, berilgan egri chiziqlar bilan chegaralangan hududni berilgan o'q atrofida aylantirish natijasida olingan qattiq jism hajmini aniqlang.

1. \(x = {\left( {y - 2} \right)^2}\), \(x\) o'qi va \(y\) o'qi bilan chegaralangan hududni \(x\) o'qi atrofida aylantiring. Yechim
2. \(\displaystyle y = \frac{1}{x}\), \(\displaystyle x = \frac{1}{2}\), \(x = 4\) va \(x\) o'qi bilan chegaralangan hududni \(y\) o'qi atrofida aylantiring. Yechim
3. \(y = 4x\) va \(y = {x^3}\) bilan chegaralangan hududni \(y\) o'qi atrofida aylantiring. Ushbu muammo uchun \(x \ge 0\) deb hisoblang. Yechim
4. \(y = 4x\) va \(y = {x^3}\) bilan chegaralangan hududni \(x\) o'qi atrofida aylantiring. Ushbu muammo uchun \(x \ge 0\) deb hisoblang. Yechim
5. \(y = 2x + 1\), \(y = 3\) va \(x = 4\) bilan chegaralangan hududni \(y = 10\) chizig‘i atrofida aylantiring. Yechim
6. \(x = {y^2} - 4\) va \(x = 6 - 3y\) bilan chegaralangan hududni \(y = - 8\) chizig‘i atrofida aylantiring. Yechim
7. \(y = {x^2} - 6x + 9\) va \(y = - {x^2} + 6x - 1\) bilan chegaralangan hududni \(x = 8\) chizig‘i atrofida aylantiring. Yechim
8. \(\displaystyle y = \frac{{{{\bf{e}}^{\frac{1}{2}x}}}}{{x + 2}}\), \(\displaystyle y = 5 - \frac{1}{4}x\), \(x = - 1\) va \(x = 6\) bilan chegaralangan hududni \(x = - 2\) chizig‘i atrofida aylantiring. Yechim