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

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

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

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

  3. Evaluate $\int\!\!\!\int_R x\,e^{y} dxdy$ over $R = \{(r\cos\theta,r\sin\theta): 0\le r\le 1, \pi \le\theta\le {{3\,\pi}\over{2}} \}$.

    $\displaystyle -\int_{0}^{1}{e^ {- 2\,r }\,\left(r\,e^{r}-r\right)\;dr}$ = $\displaystyle {{e^ {- 2 }\,\left(8\,e-3\right)}\over{4}}-{{3}\over{4}}$ $\displaystyle -\int_{0}^{1}{e^ {- 2\,r }\,\left(e^{r}-1\right)\;dr}$ = $\displaystyle {{e^ {- 2 }\,\left(2\,e-1\right)}\over{2}}-{{1}\over{2}}$ $\displaystyle -\int_{0}^{1}{e^ {- r }\,\left(e^{r}-1\right)\;dr}$ = $\displaystyle 1-e^ {- 1 }\,\left(e+1\right)$ $\displaystyle -\int_{0}^{1}{r\,e^ {- r }\,\left(e^{r}-1\right)\;dr}$ = $\displaystyle 1-{{e^ {- 1 }\,\left(e+4\right)}\over{2}}$

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

    $\displaystyle -{{\int_{0}^{3}{r^3\;dr}}\over{3}}$ = $\displaystyle -{{27}\over{4}}$ $\displaystyle {{\int_{0}^{3}{r^4\;dr}}\over{4}}$ = $\displaystyle {{243}\over{20}}$ $\displaystyle -{{\int_{0}^{3}{r^4\;dr}}\over{3}}$ = $\displaystyle -{{81}\over{5}}$ $\displaystyle {{\int_{0}^{3}{r^5\;dr}}\over{4}}$ = $\displaystyle {{243}\over{8}}$

  5. Find the Jacobian of the transformation $\displaystyle x={{u}\over{v}}$ and y = v

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

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

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

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

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

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

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

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

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

  10. Find the region $R$ bounded by the planes $x=0$,$y=0$,$z=0$, and 6z + 2y + 3x = 6.

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

  11. Evaluate $\int\!\!\!\int\!\!\!\int_Rz dxdydz$ over the region $R$ bounded by the planes $x=0$,$y=0$,$z=0$, and 3z + 3y + x = 3.

    $\displaystyle \int_{0}^{2}{\int_{0}^{2-x}{\int_{0}^{-y-{{x}\over{3}}+1}{z\;dz}\;d y}\;dx}$ = $\displaystyle {{5}\over{27}}$

    $\displaystyle \int_{0}^{2}{\int_{0}^{1-{{x}\over{3}}}{\int_{0}^{-{{y}\over{2}}-{{ x}\over{2}}+1}{z\;dz}\;dy}\;dx}$ = $\displaystyle {{49}\over{324}}$

    $\displaystyle \int_{0}^{3}{\int_{0}^{1-{{x}\over{3}}}{\int_{0}^{-y-{{x}\over{3}}+ 1}{z\;dz}\;dy}\;dx}$ = $\displaystyle {{1}\over{8}}$

    $\displaystyle \int_{0}^{3}{\int_{0}^{2-x}{\int_{0}^{-{{y}\over{2}}-{{x}\over{2}}+ 1}{z\;dz}\;dy}\;dx}$ = $\displaystyle {{5}\over{32}}$


Department of Mathematics
Last modified: 2026-06-18