Generating...                               quiz07_n13

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

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

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

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

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

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

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

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

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

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

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

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

  7. Evaluate $\int\!\!\!\int_R \left(2\,x-y\right)\,e^{2\,y+x} dxdy$ over $R = \{(r\cos\theta,r\sin\theta): 0\le r\le 1, 0 \le\theta\le {{\pi}\over{2}} \}$.

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

  8. Find the Jacobian of the transformation x = v + u and y = 2u − v

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

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

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

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

    $\displaystyle \int_{0}^{1}{\int_{0}^{1-{{x}\over{2}}}{\int_{0}^{-y-{{x}\over{2}}+ 1}{z\;dz}\;dy}\;dx}$ = $\displaystyle {{5}\over{64}}$

    $\displaystyle \int_{0}^{2}{\int_{0}^{1-{{x}\over{2}}}{\int_{0}^{-{{y}\over{2}}-x+ 1}{z\;dz}\;dy}\;dx}$ = $\displaystyle {{5}\over{48}}$

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

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


Department of Mathematics
Last modified: 2025-09-14