📝 题目
6. 1.17 求下列立体 $\Omega$ 的体积:
(1) $\Omega$ 是由曲面 $z = {xy},x + y + z = 1$ 和 $z = 0$ 围成；
(2) $\Omega$ 是由 ${y}^{2} + {z}^{2} = 1,\left| {x + y}\right| = 1,\left| {x - y}\right| = 1$ 围成.
💡 答案与解析
6. 1.12 (1) $\displaystyle{\int }_{0}^{1}\mathrm{\;d}y{\int }_{{\mathrm{e}}^{y}}^{\mathrm{e}}f\left( {x,y}\right) \mathrm{d}x$ ;
(2) $\displaystyle{\int }_{0}^{4}\mathrm{\;d}x{\int }_{\frac{x}{3}}^{\sqrt{x}}f\left( {x,y}\right) \mathrm{d}y + {\int }_{4}^{6}\mathrm{\;d}x{\int }_{\frac{x}{3}}^{2}f\left( {x,y}\right) \mathrm{d}y$ ；
(3) $\displaystyle{\int }_{-1}^{0}\mathrm{\;d}y{\int }_{-\sqrt{1 - {y}^{2}}}^{\sqrt{1 - {y}^{2}}}f\left( {x,y}\right) \mathrm{d}x$ ;
(4) $\displaystyle{\int }_{0}^{\sqrt{2}}\mathrm{\;d}y{\int }_{0}^{{y}^{2}}f\left( {x,y}\right) \mathrm{d}x + {\int }_{\sqrt{2}}^{2}\mathrm{\;d}y{\int }_{0}^{2}f\left( {x,y}\right) \mathrm{d}x$ .
6.1.13 (1) 当 $k$ 为奇数时,积分值为 0,当 $k$ 为偶数时,积分值为
$$ \frac{1}{{2}^{m - 1}} \cdot \frac{{p}^{m + k + 2}}{\left( {k + 1}\right) \left( {{2m} + k + 3}\right) }; $$
(2) ${10};\;\left( 3\right) \frac{\pi }{8}$ ; (4) $\frac{3 + {\mathrm{e}}^{2}}{4}$ ;
(5) 18; (6) $4\sin 1 - 4\sin 2 - 2\cos 2 + 2$ .
6.1.14 (1) 1; (2) $\frac{1}{2}\left( {1 - {\mathrm{e}}^{-1}}\right)$ .
6.1.17 (1) $\frac{17}{12} - 2\ln 2;\;\left( 2\right) {2\pi } - \frac{8}{3}$ .
6.1.20 证 $\left| {F\left( {\xi + {\Delta \xi },\eta + {\Delta \eta }}\right) - F\left( {\xi ,\eta }\right) }\right| \leq {2M\pi }\left\lbrack {{h}^{2} - {\left( h - \delta \right) }^{2}}\right\rbrack$ ,其中
$$ M = \mathop{\sup }\limits_{{{x}^{2} + {y}^{2} \leq {R}^{2}}}\left| {f\left( {x,y}\right) }\right| ,\;\delta = \sqrt{\Delta {\xi }^{2} + \Delta {\eta }^{2}}. $$
6.1.21 (1) $\displaystyle{\int }_{0}^{1}\mathrm{\;d}z{\int }_{0}^{z}\mathrm{\;d}y{\int }_{z - y}^{1 - y}f\left( {x,y}\right) \mathrm{d}x + {\int }_{0}^{1}\mathrm{\;d}z{\int }_{z}^{1}\mathrm{\;d}y{\int }_{0}^{1 - y}f\left( {x,y}\right) \mathrm{d}x$ ;
(2) $\displaystyle{\int }_{0}^{1}\mathrm{\;d}z{\int }_{-z}^{z}\mathrm{\;d}y{\int }_{-\sqrt{{z}^{2} - {y}^{2}}}^{\sqrt{{z}^{2} - {y}^{2}}}f\left( {x,y}\right) \mathrm{d}x$ .
6.1.22 (1) $\frac{1}{364};\;\left( 2\right) \frac{1}{2}\ln 2 - \frac{5}{16}$ ;
(3) $\frac{4\pi }{{a}^{2}}\left( {\frac{\sin {aR}}{a} - R\cos {aR}}\right)$ ; (4) $\frac{430}{21}\pi$ .