Asosiy tarkibga o'tish

6.3-bo'lim : Halqali ovoz balandligi

4. \(y = 2{x^2}\) va \(y = {x^3}\) bilan chegaralangan hududni \(x\) o'qi atrofida aylantirish natijasida olingan qattiq jismning hajmini aniqlash uchun disklar/halqalar 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 parabola \(y = 2{x^2}\) 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 narrow region between them is shaded in and a small circular arrow on the x-axis marks it as the axis of rotation.

Mana, kesishish nuqtalarini aniqlash uchun ishlatiladigan ish (bular bizga keyinroq kerak bo'ladi).

\[\begin{align*}{x^3} & = 2{x^2}\\ {x^3} - 2{x^2} & = 0\\ {x^2}\left( {x - 2} \right) & = 0\hspace{0.25in} \Rightarrow \hspace{0.25in}x = 0,\,\,\,\,\,x = 2\hspace{0.25in}\,\,\,\, \Rightarrow \,\hspace{0.25in}\,\,\,\left( {0,0} \right)\,\,\,\& \,\,\,\,\left( {2,8} \right)\end{align*}\]
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 comes to a point at the origin on the left and flares out to a wide circular face at \(x = 2\), with a cone shaped hollow scooped out along its length.

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

A sketch of the solid of revolution with a representative ring drawn in and the front half of the solid cut away so that the ring can be seen. The ring is centered on the x-axis and stands vertically inside the solid, with a hole through its middle. A sketch of the solid of revolution with a representative ring 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 ring is centered on the x-axis and stands vertically inside the solid.
3-qadam

Endi halqaning maydoni uchun formulani topishimiz kerak. Biz \(x\) o'qida joylashgan halqalardan 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 vakillik uzukining yana bir eskizi, unda biz uchun kerak bo'lgan turli miqdorlar mavjud.

A sketch of a representative ring with all of the quantities needed put in. The ring is centered on the x-axis and placed at some \(x\). The distance from the x-axis up to the curve \(y = {x^3}\) is labeled i.r. = \({x^3}\) and the distance from the x-axis up to the curve \(y = 2{x^2}\) is labeled o.r. = \(2{x^2}\).

Eskizdan ko'rinib turibdiki, halqa \(x\) o'qining markazida joylashgan va ba'zi \(x\) da joylashgan. Halqaning ichki radiusi - \(x\) o'qidan qattiq jismning ichki chetini belgilovchi egri chiziqgacha bo'lgan masofa. Halqaning tashqi radiusi - \(x\) o'qidan qattiq jismning tashqi chetini belgilovchi egri chiziqgacha bo'lgan masofa. Boshqacha aytganda,

\[{\mbox{Inner Radius}} = {x^3}\hspace{0.5in}{\mbox{Outer Radius}} = 2{x^2}\]

U holda halqaning maydoni,

\[\begin{align*}A\left( x \right) & = \pi \left[ {{{\left( {{\mbox{Outer Radius}}} \right)}^2} - {{\left( {{\mbox{Inner Radius}}} \right)}^2}} \right]\\ & = \pi \left[ {{{\left( {2{x^2}} \right)}^2} - {{\left( {{x^3}} \right)}^2}} \right] = \pi \left( {4{x^4} - {x^6}} \right)\end{align*}\]
4-qadam

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

1-bosqichdagi grafikda ko'rsatilgan kesishish nuqtalaridan biz qattiq jismdagi "birinchi" halqa \(x = 0\) da, "oxirgi" halqa esa \(x = 2\) da paydo bo'lishini ko'rishimiz mumkin. Bizning chegaralarimiz shundan iboratki: \(0 \le x \le 2\).

Keyin hajm,

\[V = \int_{0}^{2}{{\pi \left( {4{x^4} - {x^6}} \right)\,dx}} = \left. {\pi \left( {\frac{4}{5}{x^5} - \frac{1}{7}{x^7}} \right)} \right|_0^2 = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{256}}{{35}}\pi }}\]