TripleIntegrals - 8-masala yechimi | MathModel.uz

15.5-bo'lim : Uch karrali integrallar

8. $0 \le x \le 2$, $0 \le y \le 1$ tomonidan aniqlangan $xy$ tekisligida $z = 4 - xy$ ostidagi va mintaqadan yuqori mintaqa hajmini aniqlash uchun uch karrali integraldan foydalaning.

Yechim

Xo'sh, keling, nima bilan shug'ullanayotganimizni his qilishimiz uchun hajmini xohlagan hududning qisqacha eskizi bilan boshlaylik. Biz bu hududni $E$ deb ataymiz.

$A three dimensional sketch of the region $E$ drawn with the traditional axes. The orange surface across the top is the graph of $z = 4 - xy$, a gently sloping sheet that falls off a little as $x$ and $y$ increase. The blue and red surfaces are the two sides of the solid that are visible from this angle and the bottom of the region is the rectangle $0 \le x \le 2$, $0 \le y \le 1$ in the $xy$-plane, so the solid looks like a low rectangular block with a slightly tilted top.$

$A three dimensional sketch of the same region $E$, this time drawn with a set of “box” axes to make the solid easier to visualize. The tube is shaded red with the sloped yellow face on the front, and it runs out to about $x = 5$ with $z$ reaching up to roughly 2 near the $yz$-plane end.$

Biz "an'anaviy" o'qlar to'plami bilan bir qatorda "quti" o'qlari to'plami bilan eskizlarni berdik, bu sirt va mintaqani tasavvur qilishda yordam beradi.

Mintaqaning yuqori qismi (to'q sariq rangli sirt) $z = 4 - xy$ grafigidir. Ko'rsatilgan ikki tomon (ko'k va qizil yuzalar) mintaqaning yo'nalishini hisobga olgan holda biz ko'rishimiz mumkin bo'lgan mintaqaning ikki tomonini ko'rsatadi. Mintaqaning pastki qismi $xy$-samolyotidir.

2-qadam

Bu qattiq jismning hajmi quyidagicha ifodalanadi:

\[V = \iiint\limits_{E}{{dV}}\]

3-qadam

Shunday qilib, endi biz integral uchun o'rnatilgan chegaralarni olishimiz kerak. Yuqoridagi eskizdan biz $z$ uchun quyidagi limitlarga ega bo'lishini bilamiz.

\[0 \le z \le 4 - xy\]

Bizga $D$ uchun ham limitlar kerak bo'ladi. Bu holda, bu juda oddiy, masalan, biz $D$ faqat $xy$ tekisligidagi to'rtburchaklar ekanligini ko'rishimiz mumkin va aslida limitlar muammo bayonotida quyidagicha berilgan:

\[\begin{array}{c}0 \le x \le 2\\ 0 \le y \le 1\end{array}\]

Haqiqatan ham, bitta buyurtma va ikkinchisini bajarishning afzalligi yo'q, shuning uchun bu holda biz $y$ va keyin $x$ ni birlashtiramiz.

Endi, bu chegaralarning barchasini integralga ulash,

\[V = \iiint\limits_{E}{{dV}} = \int_{0}^{2}{{\int_{0}^{1}{{\int_{0}^{{4 - xy}}{{dz}}\,dy}}\,dx}}\]

4-qadam

OK, endi biz qilishimiz kerak bo'lgan narsa - bu integralni baholash. Mana $z$ integratsiyasi.

\[\begin{align*}V & = \int_{0}^{2}{{\int_{0}^{1}{{\left. z \right|_0^{4 - xy}\,dy}}\,dx}}\\ & = \int_{0}^{2}{{\int_{0}^{1}{{4 - xy\,dy}}\,dx}}\end{align*}\]

5-qadam

Endi $y$ integratsiyasini bajaramiz.

\[\begin{align*}V & = \int_{0}^{2}{{\left. {\left( {4y - \frac{1}{2}x{y^2}} \right)} \right|_0^1\,dx}}\\ & = \int_{0}^{2}{{4 - \frac{1}{2}x\,dx}}\end{align*}\]

6-qadam

Va nihoyat, mintaqa hajmini olish uchun $x$ integratsiyasini qilaylik.

\[V = \iiint\limits_{E}{{dV}} = \left. {\left( {4x - \frac{1}{4}{x^2}} \right)} \right|_0^2 = \require{bbox} \bbox[2pt,border:1px solid black]{7}\]