VolumeWithCylinder - 5-masala yechimi | MathModel.uz

6.4-bo'lim : Silindrli hajm

5. $y = 2x + 1$, $y = 3$ va $x = 4$ bilan chegaralangan hududni $y = 10$ chiziq 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 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 horizontal dashed line at $y = 10$ 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 $y = 10$. It is a solid cylinder lying on its side, running from $x = 1$ to $x = 4$, that has been hollowed out from the left end by a cone shaped cavity.$

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 dashed line $y = 10$ and lies inside 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 dashed line $y = 10$ and lies inside the solid.$

3-qadam

Endi silindrning sirt maydoni uchun formulani topishimiz kerak. Chunki biz gorizontal o'qda joylashgan silindrlardan foydalanamiz (i.e. $x$ o'qiga parallel) biz bilamizki, maydon formulasi $y$ nuqtai nazaridan bo'lishi kerak. Shuning uchun biz egri chiziqlar tenglamalarini $y$ ko'rinishida qayta yozishimiz kerak bo'ladi.

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 dashed line $y = 10$ with its lower edge at some $y$, and the distance from the axis of rotation down to that edge is labeled rad. = $10 - y$. The horizontal distance from the line $x = \frac{1}{2}\left( {y - 1} \right)$ out to $x = 4$ is labeled width = $\frac{9}{2} - \frac{1}{2}y$.$

Eskizdan biz silindrning $y = 10$ chizig'ida markazlashtirilganligini va silindrning pastki cheti qandaydir $y$ da joylashganligini ko'rishimiz mumkin.

Tsilindrning radiusi aylanish o'qidan silindrning pastki chetiga qadar bo'lgan masofadir (i.e. $10 - y$).

Tsilindrning o'ng qirrasi qattiq jismning o'ng qismini belgilovchi egri chiziqda joylashgan va $y$ o'qidan 4 masofada joylashgan. Tsilindrning chap qirrasi qattiq jismning chap qismini belgilovchi egri chiziqda joylashgan va $y$ o'qidan $\frac{1}{2}\left( {y - 1} \right)$ masofasi. Kenglik bu ikkalasining farqidir.

Shunday qilib, silindrning radiusi va kengligi,

\[{\mbox{Radius}} = 10 - y\hspace{0.25in}\hspace{0.25in}{\mbox{Width}} = 4 - \frac{1}{2}\left( {y - 1} \right) = \frac{9}{2} - \frac{1}{2}y\]

U holda silindrning maydoni,

\[A\left( y \right) = 2\pi \left( {{\mbox{Radius}}} \right)\left( {{\mbox{Height}}} \right) = 2\pi \left( {10 - y} \right)\left( {\frac{9}{2} - \frac{1}{2}y} \right) = 2\pi \left( {45 - \frac{{19}}{2}y + \frac{1}{2}{y^2}} \right)\]

4-qadam

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

1-bosqichdagi grafikdan biz qattiq jismdagi "birinchi" silindr $y = 3$ da, "oxirgi" silindr esa $y = 9$ da paydo bo'lishini ko'rishimiz mumkin. Bizning chegaralarimiz quyidagicha: $3 \le y \le 9$.

Keyin hajm,

\[V = \int_{3}^{9}{{2\pi \left( {45 - \frac{{19}}{2}y + \frac{1}{2}{y^2}} \right)\,dy}} = 2\left. {\pi \left( {45y - \frac{{19}}{4}{y^2} + \frac{1}{6}{y^3}} \right)} \right|_3^9 = \require{bbox} \bbox[2pt,border:1px solid black]{{90\pi }}\]