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6.3-bo'lim : Halqali ovoz balandligi

5. \(y = 6{{\bf{e}}^{ - 2x}}\) va \(y = 6 + 4x - 2{x^2}\) bilan chegaralangan hududni \(x = 0\) va \(x = 1\) o‘rtasida \(y = - 2\) 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 parabola \(y = 6 + 4x - 2{x^2}\) is the upper boundary and the curve \(y = 6{{\bf{e}}^{ - 2x}}\) is the lower boundary, and the two meet at \(\left( {0,6} \right)\). The region between them runs from \(x = 0\) to the vertical line \(x = 1\) and is shaded in. A horizontal dashed line at \(y = - 2\) with a small circular arrow on it marks the axis of rotation.

Chapdagi kesishish nuqtasi uchun \(x = 0\) ni ikkala tenglamaga ulash orqali tezkor tekshirish, biz grafikdan taxmin qilganimizdek kesishish nuqtasi aslida \(\left( {0,6} \right)\) ekanligini ko'rsatadi. Bizga bu nuqta birozdan keyin kerak bo'ladi. Mintaqaning eskizidan o'ng tomonda kesishish nuqtasi yo'qligi ham aniq.

2-qadam

Bu erda inqilobning qattiq qismining eskizi.

A sketch of the solid of revolution obtained by rotating the region about the line \(y = - 2\). It is a barrel shaped solid lying on its side that is wide at the left end and narrows towards the right, with a hole running through the middle along the axis of rotation.

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 \(y = - 2\) and stands vertically 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 \(y = - 2\) and stands vertically inside the solid.
3-qadam

Endi halqaning maydoni uchun formulani topishimiz kerak. Chunki biz gorizontal o'qda joylashgan halqalardan foydalanamiz (i.e. \(x\) o'qiga parallel) biz bilamizki, maydon formulasi \(x\) nuqtai nazaridan bo'lishi kerak. Shuning uchun, egri chiziqlar tenglamalari \(x\) (bu holda ular allaqachon mavjud) jihatidan bo'lishi kerak.

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 \(y = - 2\) and placed at some \(x\). The inner radius is labeled i.r. = \(2 + 6{{\bf{e}}^{ - 2x}}\), made up of the distance 2 from the axis of rotation up to the x-axis plus the distance \(6{{\bf{e}}^{ - 2x}}\) from the x-axis up to the lower curve. The outer radius is labeled o.r. = \(8 + 4x - 2{x^2}\), made up of the same distance 2 plus the distance \(6 + 4x - 2{x^2}\) up to the upper curve.

Eskizdan biz halqaning \(y = - 2\) chizig'ida markazlashtirilganligini va qandaydir \(x\)da joylashganini ko'rishimiz mumkin.

Halqaning ichki radiusi - bu aylanish o'qidan \(x\) o'qigacha bo'lgan masofa (masofa 2), undan keyin \(x\) o'qidan qattiq jismning ichki chetini belgilovchi egri chiziqgacha bo'lgan masofa (\(6{{\bf{e}}^{ - 2x}}\) masofasi).

Xuddi shunday, halqaning tashqi radiusi aylanish o'qidan \(x\) o'qigacha bo'lgan masofa (yana 2 masofa), undan keyin \(x\) o'qidan qattiq jismning tashqi chetini belgilovchi egri chiziqgacha bo'lgan masofa (\(6 + 4x - 2{x^2}\) masofasi).

Shunday qilib, ichki va tashqi radiuslar,

\[{\mbox{Inner Radius}} = 2 + 6{{\bf{e}}^{ - 2x}}\hspace{0.5in}{\mbox{Outer Radius}} = 2 + 6 + 4x - 2{x^2} = 8 + 4x - 2{x^2}\]

U holda halqaning maydoni,

\[\begin{align*}A\left( x \right) & = \pi \left[ {{{\left( {{\mbox{Outer Radius}}} \right)}^2} - {{\left( {{\mbox{Inner Radius}}} \right)}^2}} \right]\\ & = \pi \left[ {{{\left( {8 + 4x - 2{x^2}} \right)}^2} - {{\left( {2 + 6{{\bf{e}}^{ - 2x}}} \right)}^2}} \right]\\ & = \pi \left( {60 + 64x - 16{x^2} - 16{x^3} + 4{x^4} - 24{{\bf{e}}^{ - 2x}} - 36{{\bf{e}}^{ - 4x}}} \right)\end{align*}\]
4-qadam

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

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

Keyin hajm,

\[\begin{align*}V & = \int_{0}^{1}{{\pi \left( {60 + 64x - 16{x^2} - 16{x^3} + 4{x^4} - 24{{\bf{e}}^{ - 2x}} - 36{{\bf{e}}^{ - 4x}}} \right)\,dx}}\\ & = \left. {\pi \left( {60x + 32{x^2} - \frac{{16}}{3}{x^3} - 4{x^4} + \frac{4}{5}{x^5} + 12{{\bf{e}}^{ - 2x}} + 9{{\bf{e}}^{ - 4x}}} \right)} \right|_0^1 = \require{bbox} \bbox[2pt,border:1px solid black]{{\left( {\frac{{937}}{{15}} + 12{{\bf{e}}^{ - 2}} + 9{{\bf{e}}^{ - 4}}} \right)\pi }}\end{align*}\]