VolumeWithCylinder - 1-masala yechimi | MathModel.uz

6.4-bo'lim : Silindrli hajm

1. $x = {\left( {y - 2} \right)^2}$, $x$ o'qi va $y$ o'qi bilan chegaralangan hududni $x$ o'qi atrofida aylantirish natijasida olingan qattiq jismning hajmini silindrlar usulidan foydalaning.

Yechim

Muammoni biror joyda boshlashimiz kerak, shuning uchun "oddiy" ni boshlaylik.

Chegaralangan mintaqaning qanday ko'rinishini bilish ushbu turdagi muammolarning ko'pchiligiga yordam beradi, chunki biz maydon formulasini o'rnatish uchun barcha egri chiziqlar bir-biriga qanday bog'liqligini bilishimiz kerak va bizga grafik ko'pincha yordam beradigan integral uchun limitlar kerak bo'ladi.

Bu erda aylanish o'qi bilan chegaralangan hududning eskizi ko'rsatilgan.

$A sketch of the bounded region with the axis of rotation shown. The lower half of the parabola $x = {\left( {y - 2} \right)^2}$ runs from $\left( {0,2} \right)$ down to $\left( {4,0} \right)$ and the shaded region is the area under it, bounded above by the curve, on the left by the y-axis and below by the x-axis. A small circular arrow on the x-axis marks it as the axis of rotation.$

E'tibor bering, biz bu erda parabolaning faqat pastki yarmidan foydalandik, chunki agar biz yuqori yarmini ham qo'shsak, uning ustidagi mintaqani bog'laydigan hech narsa bo'lmaydi. Shuning uchun, $x$ o'qi va $y$ o'qi chegaralovchi egri chiziqlar bo'lishi uchun biz parabolaning pastki yarmidan past bo'lgan qismdan foydalanishimiz kerak.

2-qadam

Bu erda inqilobning qattiq qismining eskizi.

$A sketch of the solid of revolution obtained by rotating the region about the x-axis. It is a shape rather like a horn, wide and rounded at the left end where it meets the y-axis and tapering to a point at $x = 4$ on the x-axis.$

Bu erda vakillik silindrining bir nechta eskizlari. Chapdagi rasmda qattiq jismning old yarmi kesilgan vakili silindr ko'rsatilgan va o'ngdagi rasmda qattiq jismning orqa yarmining "simli ramkasi" bo'lgan vakili silindr ko'rsatilgan (i.e. qattiq jismning orqa yarmining chetlarini ifodalovchi egri chiziqlar).

A sketch of the solid of revolution with a representative cylinder drawn in and the front half of the solid cut away so that the cylinder can be seen. The cylinder is centered on the x-axis, lies inside the horn shaped solid near the wide left end, and its edges run from the y-axis out to the curve that forms the edge of the solid.

A sketch of the solid of revolution with a representative cylinder drawn in and only a “wire frame” of the back half of the solid shown, so just the curves that form the edges of the solid are drawn. The cylinder is centered on the x-axis and lies inside the solid near the wide left end.

3-qadam

Endi silindrning sirt maydoni uchun formulani topishimiz kerak. Biz $x$ o'qida joylashgan silindrlardan foydalanayotganimiz sababli, maydon formulasi $y$ nuqtai nazaridan bo'lishi kerakligini bilamiz. Shuning uchun, egri chiziqlar tenglamasi $y$ nuqtai nazaridan bo'lishi kerak (bu holda ular allaqachon mavjud).

Mana, biz kiritishimiz kerak bo'lgan turli miqdorlar bilan vakili silindrning yana bir eskizi.

$A sketch of a representative cylinder with all of the quantities needed put in. The cylinder is centered on the x-axis with its upper edge at some $y$, and the distance from the x-axis up to that edge is labeled radius = $y$. The distance from the y-axis across to the curve $x = {\left( {y - 2} \right)^2}$ is labeled width = ${\left( {y - 2} \right)^2}$.$

Eskizdan biz silindrning $x$ o'qida joylashganligini va silindrning yuqori qirrasi qandaydir $y$ da joylashganligini ko'rishimiz mumkin.

Tsilindrning radiusi - bu $x$ o'qidan silindrning yuqori chetigacha bo'lgan masofa (i.e. $y$). Tsilindrning kengligi $y$ o'qidan qattiq jismning chetini belgilovchi egri chiziqgacha bo'lgan masofa (${\left( {y - 2} \right)^2}$ masofasi).

Shunday qilib, silindrning radiusi va kengligi,

\[{\mbox{Radius}} = y\hspace{0.25in}\hspace{0.25in}{\mbox{Width}} = {\left( {y - 2} \right)^2}\]

U holda silindrning maydoni,

\[A\left( y \right) = 2\pi \left( {{\mbox{Radius}}} \right)\left( {{\mbox{Width}}} \right) = 2\pi \left( y \right){\left( {y - 2} \right)^2} = 2\pi \left( {4y - 4{y^2} + {y^3}} \right)\]

4-qadam

Yakuniy bosqich - bu hajm uchun integralni o'rnatish va uni baholash.

1-bosqichdagi grafikdan biz qattiq jismdagi "birinchi" silindr $y = 0$ da, "oxirgi" silindr esa $y = 2$ da paydo bo'lishini ko'rishimiz mumkin. Bizning chegaralarimiz quyidagicha: $0 \le y \le 2$.

Keyin hajm,

\[V = \int_{0}^{2}{{2\pi \left( {4y - 4{y^2} + {y^3}} \right)\,dy}} = 2\left. {\pi \left( {2{y^2} - \frac{4}{3}{y^3} + \frac{1}{4}{y^4}} \right)} \right|_0^2 = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{8}{3}\pi }}\]