Asosiy tarkibga o'tish

6.3-bo'lim : Halqali ovoz balandligi

8. \(y = 2x + 1\), \(x = 4\) va \(y = 3\) bilan chegaralangan hududni \(x = - 4\) chizig‘i atrofida aylantirish natijasida olingan qattiq jism hajmini 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 region is a triangle with corners at \(\left( {1,3} \right)\), \(\left( {4,3} \right)\) and \(\left( {4,9} \right)\), bounded above by the line \(y = 2x + 1\), below by the line \(y = 3\) and on the right by the line \(x = 4\), and it is shaded in. A vertical dashed line at \(x = - 4\) with a small circular arrow on it marks the axis of rotation.
2-qadam

Bu erda inqilobning qattiq qismining eskizi.

A sketch of the solid of revolution obtained by rotating the region about the line \(x = - 4\). It is a wide, shallow bowl shaped solid centered on the axis of rotation, open at the top with sloping inner walls.

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 dashed line \(x = - 4\) and lies horizontally inside the solid. 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 dashed line \(x = - 4\) and lies horizontally inside the solid.
3-qadam

Endi halqaning maydoni uchun formulani topishimiz kerak. Chunki biz vertikal o'qda joylashgan halqalardan foydalanamiz (i.e. \(y\) o'qiga parallel) biz bilamizki, maydon formulasi \(y\) nuqtai nazaridan bo'lishi kerak. Shuning uchun, egri chiziqlar tenglamalari \(y\) bo'yicha bo'lishi kerak va shuning uchun biz \(y\) nuqtai nazaridan chiziq tenglamasini qayta yozishimiz kerak bo'ladi.

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 dashed line \(x = - 4\) and placed at some \(y\). The inner radius is labeled i.r. = \(\frac{1}{2}y + \frac{7}{2}\), made up of the distance 4 from the axis of rotation to the y-axis plus the distance \(\frac{1}{2}\left( {y - 1} \right)\) out to the line \(x = \frac{1}{2}\left( {y - 1} \right)\). The outer radius is labeled o.r. = 8, made up of the same distance 4 plus the distance 4 out to the line \(x = 4\).

Eskizdan biz halqaning \(x = - 4\) chizig'ida markazlashtirilganligini va qandaydir \(y\)da joylashganini ko'rishimiz mumkin.

Halqaning ichki radiusi - bu aylanish o'qidan \(y\) o'qigacha bo'lgan masofa (4 masofa), undan keyin \(y\) o'qidan qattiq jismning ichki chetini belgilovchi egri chiziqgacha bo'lgan masofa (\(\frac{1}{2}\left( {y - 1} \right)\) masofasi).

Xuddi shunday, halqaning tashqi radiusi aylanish o'qidan \(y\) o'qigacha bo'lgan masofa (yana 4 masofa) va undan keyin \(y\) o'qidan qattiq jismning tashqi chetini belgilovchi egri chiziqgacha bo'lgan masofa (4 masofa).

Shunday qilib, ichki va tashqi radiuslar,

\[{\mbox{Inner Radius}} = 4 + \frac{1}{2}\left( {y - 1} \right) = \frac{1}{2}y + \frac{7}{2}\hspace{0.5in}{\mbox{Outer Radius}} = 4 + 4 = 8\]

U holda halqaning maydoni,

\[\begin{align*}A\left( x \right) & = \pi \left[ {{{\left( {{\mbox{Outer Radius}}} \right)}^2} - {{\left( {{\mbox{Inner Radius}}} \right)}^2}} \right]\\ & = \pi \left[ {{{\left( 8 \right)}^2} - {{\left( {\frac{1}{2}y + \frac{7}{2}} \right)}^2}} \right] = \pi \left( {\frac{{207}}{4} - \frac{7}{2}y - \frac{1}{4}{y^2}} \right)\end{align*}\]
4-qadam

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

1-bosqichda topilgan ikkita egri chiziqning kesishish nuqtalaridan biz qattiq jismdagi “birinchi” halqa \(y = 3\) da, “oxirgi” halqa esa \(y = 9\) da paydo bo‘lishini ko‘rishimiz mumkin. Bizning chegaralarimiz quyidagicha: \(3 \le y \le 9\).

Keyin hajm,

\[V = \int_{3}^{9}{{\pi \left( {\frac{{207}}{4} - \frac{7}{2}y - \frac{1}{4}{y^2}} \right)\,dy}} = \left. {\pi \left( {\frac{{207}}{4}y - \frac{7}{4}{y^2} - \frac{1}{{12}}{y^3}} \right)} \right|_3^9 = \require{bbox} \bbox[2pt,border:1px solid black]{{126\pi }}\]