8.3-bo'lim : Massa markazi
2. \(y = 3 - {{\bf{e}}^{ - x}}\), \(x\) o'qi, \(x = 2\) va \(y\) o'qi bilan chegaralangan mintaqa uchun massa markazini toping.
Yechim
Keling, nuqta bilan ko'rsatilgan massa markazi bilan mintaqaning qisqacha chizmasidan boshlaylik (bu nuqtaning koordinatalari, albatta, oxirgi bosqichda aniqlanishi kerak ....).
Bizga ushbu hududning maydoni ham kerak bo'ladi, shuning uchun avval uni topamiz.
\[A = \int_{0}^{2}{{3 - {{\bf{e}}^{ - x}}\,dx}} = \left. {\left( {3x + {{\bf{e}}^{ - x}}} \right)} \right|_0^2 = 5 + {{\bf{e}}^{ - 2}}\]2-qadam
Keyinchalik, ikkita momentni hisoblashimiz kerak. Quyidagi hisob-kitoblarga zichlikni kiritmadik, chunki u faqat oxirgi bosqichda bekor qilinadi.
\[\begin{align*}{M_x} & = \int_{0}^{2}{{\frac{1}{2}{{\left( {3 - {{\bf{e}}^{ - x}}} \right)}^2}\,dx}} = \int_{0}^{2}{{\frac{1}{2}\left( {9 - 6{{\bf{e}}^{ - x}} + {{\bf{e}}^{ - 2x}}} \right)\,dx}} = \left. {\frac{1}{2}\left( {9x + 6{{\bf{e}}^{ - x}} - \frac{1}{2}{{\bf{e}}^{ - 2x}}} \right)} \right|_0^2 = \frac{{25}}{4} + 3{{\bf{e}}^{ - 2}} - \frac{1}{4}{{\bf{e}}^{ - 4}}\\ {M_y} & = \int_{0}^{2}{{x\left( {3 - {{\bf{e}}^{ - x}}} \right)\,dx}} = \int_{0}^{2}{{3x - x\,{{\bf{e}}^{ - x}}\,dx}} = \left. {\left( {\frac{3}{2}{x^2} + x{{\bf{e}}^{ - x}} + {{\bf{e}}^{ - x}}} \right)} \right|_0^2 = 5 + 3{{\bf{e}}^{ - 2}}\end{align*}\]\({M_y}\) integratsiyasining ikkinchi muddati uchun biz qismlar bo'yicha quyidagi integratsiyadan foydalandik.
\[\begin{align*} & \int{{x\,{{\bf{e}}^{ - x}}\,dx}}\hspace{0.25in}\hspace{0.25in}u = x\,\,\,\,\,\,du = dx\hspace{0.25in}\hspace{0.25in}dv = {{\bf{e}}^{ - x}}dx\,\,\,\,\,v = - {{\bf{e}}^{ - x}}\\ & \int{{x\,{{\bf{e}}^{ - x}}\,dx}} = - x{{\bf{e}}^{ - x}} + \int{{{{\bf{e}}^{ - x}}\,dx}} = - x{{\bf{e}}^{ - x}} - {{\bf{e}}^{ - x}} = - \left( {x{{\bf{e}}^{ - x}} + {{\bf{e}}^{ - x}}} \right)\end{align*}\]Bu erda minus belgisi to'liq integralda atama oldidagi minus belgisi bilan bekor qilindi.
Qismlar bo'yicha integratsiyani unutmang! Bu bunday muammolar uchun juda keng tarqalgan integratsiya texnikasi.
3-qadam
Nihoyat, massa markazining koordinatalari,
\[\overline{x} = \frac{{{M_y}}}{M} = \frac{{\rho \left( {5 + 3{{\bf{e}}^{ - 2}}} \right)}}{{\rho \left( {5 + {{\bf{e}}^{ - 2}}} \right)}} = 1.05271\hspace{0.25in}\hspace{0.25in}\overline{y} = \frac{{{M_x}}}{M} = \frac{{\rho \left( {\frac{{25}}{4} + 3{{\bf{e}}^{ - 2}} - \frac{1}{4}{{\bf{e}}^{ - 4}}} \right)}}{{\rho \left( {5 + {{\bf{e}}^{ - 2}}} \right)}} = 1.29523\]U holda massa markazi: \(\require{bbox} \bbox[2pt,border:1px solid black]{{\left( {1.05271,\,\,1.29523} \right)}}\).