Asosiy tarkibga o'tish

6.4-bo'lim : Silindrli hajm

3. \(y = 4x\) va \(y = {x^3}\) bilan chegaralangan hududni \(y\) o'qi atrofida aylantirish natijasida olingan qattiq jismning hajmini silindrlar usulidan foydalaning. Ushbu muammo uchun \(x \ge 0\) deb hisoblang.

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 line \(y = 4x\) is the upper boundary and the curve \(y = {x^3}\) is the lower boundary, and the two meet at the origin and again at \(\left( {2,8} \right)\), which is marked and labeled. The region between them is shaded in and a small circular arrow on the y-axis marks it as the axis of rotation.

Grafikda ko'rsatilgan kesishish nuqtalarini olish uchun bizga birozdan keyin kerak bo'ladigan tenglamalarni bir-biriga tenglashtirish va yechish kerak.

\[\begin{align*}{x^3} & = 4x\\ {x^3} - 4x & = 0\\ x\left( {{x^2} - 4} \right) & = 0\hspace{0.25in} \Rightarrow \hspace{0.25in}x = 0,\,\,\,\,\,\,x = \pm 2\hspace{0.25in} \Rightarrow \hspace{0.25in}\left( {0,0} \right)\,\,\,\,\& \,\,\,\,\left( {2,8} \right)\end{align*}\]

E'tibor bering, muammo bayonotida \(x \ge 0\) deb taxmin qilingan va shuning uchun biz \(x = - 2\) kesishish nuqtasidan foydalanmaymiz.

2-qadam

Bu erda inqilobning qattiq qismining eskizi.

A sketch of the solid of revolution obtained by rotating the region about the y-axis. It is a bowl shaped solid, open at the top, whose rim is at a height of about 8 and whose bottom comes to a rounded point at the origin.

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 bowl shaped 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 y-axis and stands inside the bowl. A sketch of the bowl shaped 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 y-axis and stands inside the bowl.
3-qadam

Endi silindrning sirt maydoni uchun formulani topishimiz kerak. Biz \(y\) o'qida joylashgan silindrlardan foydalanayotganimiz sababli, maydon formulasi \(x\) nuqtai nazaridan bo'lishi kerakligini bilamiz. Shuning uchun, egri chiziqlar tenglamasi \(x\) 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 y-axis with its right edge at some \(x\), and the distance from the y-axis across to that edge is labeled radius = \(x\). The vertical distance from the curve \(y = {x^3}\) up to the line \(y = 4x\) is labeled height = \(4x - {x^3}\), with the two individual distances \(4x\) and \({x^3}\) also marked.

Eskizdan biz tsilindr \(y\) o'qida joylashganligini va silindrning o'ng qirrasi qandaydir \(x\) da joylashganligini ko'rishimiz mumkin.

Tsilindrning radiusi - bu \(y\) o'qidan silindrning o'ng chetigacha bo'lgan masofa (i.e. \(x\)).

Tsilindrning ustki qismi qattiq jismning yuqori qismini belgilovchi egri chiziq ustida joylashgan va \(x\) o'qidan \(4x\) masofasida joylashgan. Tsilindrning pastki qismi qattiq jismning pastki qismini belgilaydigan egri chiziqda joylashgan va \(x\) o'qidan \({x^3}\) masofasida joylashgan. Keyin balandlik bu ikkalasining farqidir.

Shunday qilib, silindrning radiusi va balandligi,

\[{\mbox{Radius}} = x\hspace{0.25in}\hspace{0.25in}{\mbox{Height}} = 4x - {x^3}\]

U holda silindrning maydoni,

\[A\left( x \right) = 2\pi \left( {{\mbox{Radius}}} \right)\left( {{\mbox{Height}}} \right) = 2\pi \left( x \right)\left( {4x - {x^3}} \right) = 2\pi \left( {4{x^2} - {x^4}} \right)\]
4-qadam

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

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

Keyin hajm,

\[V = \int_{0}^{2}{{2\pi \left( {4{x^2} - {x^4}} \right)\,dx}} = 2\left. {\pi \left( {\frac{4}{3}{x^3} - \frac{1}{5}{x^5}} \right)} \right|_0^2 = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{128}}{{15}}\pi }}\]