Generating...                               quiz07_n12

  1. Find the region $R$ bounded by the planes $x=0$,$y=0$,$z=0$, and 9z + 3y + 3x = 9.

    $R = \{(x,y,z): 0 \le x\le 3, 0 \le y\le 2-2\,x, 0 \le z\le -{{y}\over{2}}-x+1 \}$. $R = \{(x,y,z): 0 \le x\le 1, 0 \le y\le 3-x, 0 \le z\le -{{y}\over{2}}-x+1 \}$. $R = \{(x,y,z): 0 \le x\le 3, 0 \le y\le 3-x, 0 \le z\le -{{y}\over{3}}-{{x}\over{3}}+1 \}$. $R = \{(x,y,z): 0 \le x\le 1, 0 \le y\le 2-2\,x, 0 \le z\le -{{y}\over{3}}-{{x}\over{3}}+1 \}$.

  2. Evaluate $\int\!\!\!\int_R x\,y dxdy$ over $R = \{( {{u}\over{v}}, v): 1 \le u\le 4, u \le v\le 2\,u \}$.

    $\displaystyle \int_{1}^{4}{u\;du}$ = $\displaystyle {{15}\over{2}}$ $\displaystyle \int_{1}^{4}{u^2\;du}$ = 21 $\displaystyle \int_{1}^{4}{u\,\ln \left(2\,u\right)-u\,\ln u\;du}$ = $\displaystyle {{15\,\ln 2}\over{2}}$ $\displaystyle {{3\,\int_{1}^{4}{u^2\;du}}\over{2}}$ = $\displaystyle {{63}\over{2}}$

  3. Find the equivalent expression of the region for $R = \{(x,y): -1 \le x\le 1, -\sqrt{1-x^2} \le y\le 0 \}$.

    $R = \{(r\cos\theta,r\sin\theta): 0\le r\le 1, \pi \le\theta\le 2\,\pi \}$. $R = \{(r\cos\theta,r\sin\theta): 0\le r\le 1, 0 \le\theta\le \pi \}$. $R = \{(r\cos\theta,r\sin\theta): 0\le r\le 1, 0 \le\theta\le {{\pi}\over{2}} \}$.

  4. Find the surface area for $z = x\,y+1$ above $R = \{(r\cos\theta,r\sin\theta): 0\le r\le 1, 0 \le\theta\le 2\,\pi \}$.

    $\displaystyle 2\,\pi\,\int_{0}^{1}{r\,\sqrt{r^2+1}\;dr}$ = $\displaystyle 2\,\left({{2^{{{3}\over{2}}}}\over{3}}-{{1}\over{3}}\right)\,\pi$ $\displaystyle 2\,\pi\,\int_{0}^{1}{r\,\sqrt{4\,r^2+1}\;dr}$ = $\displaystyle 2\,\left({{5^{{{3}\over{2}}}}\over{12}}-{{1}\over{12}}\right)\,\pi$ $\displaystyle 2\,\pi\,\int_{0}^{1}{r\;dr}$ = $\displaystyle \pi$ $\displaystyle 2\,\pi\,\int_{0}^{1}{r^3\;dr}$ = $\displaystyle {{\pi}\over{2}}$

  5. Evaluate $\int\!\!\!\int_R \left(y+x\right)\,e^{x-y} dxdy$ over $R = \{( v+u, u-v): 0 \le u\le 1, 0 \le v\le 1 \}$.

    $\displaystyle 2\,\int_{0}^{1}{e^{2\,v}\;dv}$ = $\displaystyle 2\,\left({{e^2}\over{2}}-{{1}\over{2}}\right)$ $\displaystyle \int_{0}^{1}{e^{2\,v}\;dv}$ = $\displaystyle {{e^2}\over{2}}-{{1}\over{2}}$ $\displaystyle \left(2\,e-2\right)\,\int_{0}^{1}{e^ {- v }\;dv}$ = $\displaystyle \left(1-e^ {- 1 }\right)\,\left(2\,e-2\right)$ $\displaystyle \left(e-1\right)\,\int_{0}^{1}{e^ {- v }\;dv}$ = $\displaystyle \left(1-e^ {- 1 }\right)\,\left(e-1\right)$

  6. Evaluate $\int\!\!\!\int\!\!\!\int_Rz dxdydz$ over the region $R$ bounded by the planes $x=0$,$y=0$,$z=0$, and 2z + 2y + x = 2.

    $\displaystyle \int_{0}^{2}{\int_{0}^{1-{{x}\over{2}}}{\int_{0}^{-y-{{x}\over{2}}+ 1}{z\;dz}\;dy}\;dx}$ = $\displaystyle {{1}\over{12}}$

    $\displaystyle \int_{0}^{3}{\int_{0}^{2-{{2\,x}\over{3}}}{\int_{0}^{-y-{{x}\over{2 }}+1}{z\;dz}\;dy}\;dx}$ = $\displaystyle {{5}\over{16}}$

    $\displaystyle \int_{0}^{2}{\int_{0}^{2-{{2\,x}\over{3}}}{\int_{0}^{-{{y}\over{2}} -{{x}\over{3}}+1}{z\;dz}\;dy}\;dx}$ = $\displaystyle {{20}\over{81}}$

    $\displaystyle \int_{0}^{3}{\int_{0}^{1-{{x}\over{2}}}{\int_{0}^{-{{y}\over{2}}-{{ x}\over{3}}+1}{z\;dz}\;dy}\;dx}$ = $\displaystyle {{49}\over{256}}$

  7. Evaluate $\int\!\!\!\int_R x\,y^3 dxdy$ over $R = \{(r\cos\theta,r\sin\theta): 0\le r\le 2, 0 \le\theta\le {{\pi}\over{2}} \}$.

    $\displaystyle {{\int_{0}^{2}{r^5\;dr}}\over{4}}$ = $\displaystyle {{8}\over{3}}$ $\displaystyle {{\int_{0}^{2}{r^4\;dr}}\over{4}}$ = $\displaystyle {{8}\over{5}}$ $\displaystyle {{\int_{0}^{2}{r^2\;dr}}\over{2}}$ = $\displaystyle {{4}\over{3}}$ $\displaystyle {{\int_{0}^{2}{r^3\;dr}}\over{2}}$ = 2

  8. Evaluate $\displaystyle \int_{0}^{1}{\int_{0}^{\sqrt{1-x^2}}{e^{-y^2-x^2}\;dy}\;dx}$

    $\displaystyle {{\pi\,\int_{0}^{1}{e^ {- r }\;dr}}\over{2}}$ = $\displaystyle {{\left(1-e^ {- 1 }\right)\,\pi}\over{2}}$ $\displaystyle {{\pi\,\int_{0}^{1}{r\,e^ {- r^2 }\;dr}}\over{2}}$ = $\displaystyle {{\left({{1}\over{2}}-{{e^ {- 1 }}\over{2}}\right)\,\pi}\over{2}}$ $\displaystyle \pi\,\int_{0}^{1}{e^ {- r }\;dr}$ = $\displaystyle \left(1-e^ {- 1 }\right)\,\pi$ $\displaystyle \pi\,\int_{0}^{1}{r\,e^ {- r^2 }\;dr}$ = $\displaystyle \left({{1}\over{2}}-{{e^ {- 1 }}\over{2}}\right)\,\pi$

  9. Find the Jacobian of the transformation $\displaystyle x=u\,v+u^2$ and $\displaystyle y=u\,v^2$

    $\displaystyle {{1}\over{v}}$ −3 u $\displaystyle u\,v^2+4\,u^2\,v$

  10. Find the surface area for $z = 2\,y+x^2$ above $R = \{(x,y): 0 \le x\le 1, 0 \le y\le x \}$.

    $\displaystyle \int_{0}^{1}{x^3+x^2\;dx}$ = $\displaystyle {{7}\over{12}}$ $\displaystyle 2\,\int_{0}^{1}{x^2\;dx}$ = $\displaystyle {{2}\over{3}}$ $\displaystyle \sqrt{6}\,\int_{0}^{1}{x\;dx}$ = $\displaystyle {{\sqrt{6}}\over{2}}$ $\displaystyle \int_{0}^{1}{x\,\sqrt{4\,x^2+5}\;dx}$ = $\displaystyle -{{5^{{{3}\over{2}}}-27}\over{12}}$


Department of Mathematics
Last modified: 2025-06-19