Generating...                               quiz06c_n6

  1. Evaluate $\displaystyle \int_{0}^{{{\pi}\over{4}}}{\int_{0}^{{{\pi}\over{4}}}{\sin \left(y+ x\right)\;dx}\;dy}$.

    $\displaystyle \int_{0}^{{{\pi}\over{4}}}{\cos y-\cos \left({{4\,y+\pi}\over{4}} \right)\;dy}$ = $\displaystyle \sqrt{2}-1$ $\displaystyle \int_{0}^{{{\pi}\over{4}}}{\cos y\;dy}$ = $\displaystyle {{1}\over{\sqrt{2}}}$ $\displaystyle \left(1-{{1}\over{\sqrt{2}}}\right)\,\int_{0}^{{{\pi}\over{4}}}{ \sin y\;dy}$ = $\displaystyle \left(1-{{1}\over{\sqrt{2}}}\right)^2$ $\displaystyle \int_{0}^{{{\pi}\over{4}}}{\sin y\;dy}$ = $\displaystyle 1-{{1}\over{\sqrt{2}}}$

  2. Evaluate $\displaystyle \int_{0}^{1}{\int_{1}^{4}{{{y-x}\over{\sqrt{y}}}\;dy}\;dx}$.

    $\displaystyle \int_{0}^{1}{{{\left(4-x\right)^2}\over{4}}-\left(1-x\right)^2\;dx}$ = $\displaystyle {{11}\over{3}}$ $\displaystyle -{{\int_{0}^{1}{6\,x-14\;dx}}\over{3}}$ = $\displaystyle {{11}\over{3}}$ $\displaystyle -2\,\int_{0}^{1}{x\;dx}$ = $\displaystyle {{11}\over{3}}$ $\displaystyle \int_{0}^{1}{{{2\,x-1}\over{2}}-4\,x+8\;dx}$ = $\displaystyle {{11}\over{3}}$

  3. Evaluate $\displaystyle\int\!\!\int_D 2\,x\,y dxdy$ where $D$ is the region bounded by $\displaystyle x=y^2$ and x = y.

    $\displaystyle \int_{0}^{1}{y\,\left({{y^2}\over{2}}-{{y^4}\over{2}}\right)\;dy}$ = $\displaystyle {{1}\over{24}}$ $\displaystyle 2\,\int_{0}^{1}{y\,\left({{y^2}\over{2}}-{{y^4}\over{2}}\right)\;dy }$ = $\displaystyle {{1}\over{12}}$ $\displaystyle \int_{1}^{2}{y\,\left({{y^2}\over{2}}-{{y^4}\over{2}}\right)\;dy}$ = $\displaystyle -{{27}\over{8}}$ $\displaystyle 2\,\int_{1}^{2}{y\,\left({{y^2}\over{2}}-{{y^4}\over{2}}\right)\;dy }$ = $\displaystyle -{{27}\over{4}}$

  4. Evaluate $\displaystyle \int_{0}^{1}{\int_{0}^{1}{\left(y+3\,x\right)^3\;dx}\;dy}$.

    $\displaystyle {{\int_{0}^{1}{4\,y^3+18\,y^2+36\,y+27\;dy}}\over{4}}$ = 13 $\displaystyle \int_{0}^{1}{\left(y+3\right)^3\;dy}$ = $\displaystyle {{175}\over{4}}$ $\displaystyle \int_{0}^{1}{\left(y+3\right)^4\;dy}$ = $\displaystyle {{781}\over{5}}$ $\displaystyle \int_{0}^{1}{{{y^3+9\,y^2+27\,y+27}\over{3}}-{{y^3}\over{3}}\;dy}$ = $\displaystyle {{29}\over{2}}$

  5. Evaluate $\displaystyle \int_{0}^{{{\pi}\over{2}}}{\int_{0}^{\pi}{-2\,\sin y-3\,\cos x\;dy} \;dx}$.

    $\displaystyle -2\,\pi\,\int_{0}^{{{\pi}\over{2}}}{\sin x\;dx}$ = $\displaystyle -5\,\pi$ $\displaystyle \int_{0}^{{{\pi}\over{2}}}{3\,\pi\,\cos x-4\;dx}$ = $\displaystyle -5\,\pi$ $\displaystyle \int_{0}^{{{\pi}\over{2}}}{-3\,\pi\,\cos x-4\;dx}$ = $\displaystyle -5\,\pi$ $\displaystyle 2\,\pi\,\int_{0}^{{{\pi}\over{2}}}{\sin x\;dx}$ = $\displaystyle -5\,\pi$

  6. Evaluate $\displaystyle \int_{0}^{1}{\int_{-3}^{-2}{e^{x+3}\,y+x\;dx}\;dy}$.

    $\displaystyle \int_{0}^{1}{e^2\,y-e\,y+1\;dy}$ = $\displaystyle {{e-6}\over{2}}$ $\displaystyle \int_{0}^{1}{3\,e^ {- 2 }\,y-3\,e^ {- 3 }\,y+1\;dy}$ = $\displaystyle {{e-6}\over{2}}$ $\displaystyle {{\int_{0}^{1}{\left(2\,e-2\right)\,y-5\;dy}}\over{2}}$ = $\displaystyle {{e-6}\over{2}}$ $\displaystyle \int_{0}^{1}{e^ {- 2 }\,y-e^ {- 3 }\,y+1\;dy}$ = $\displaystyle {{e-6}\over{2}}$

  7. Evaluate $\displaystyle \int_{1}^{2}{y\,\int_{1}^{2}{{{x}\over{\sqrt{y^2+x^2}}}\;dx}\;dy}$.

    $\displaystyle \int_{1}^{2}{y\,\left({{2}\over{\sqrt{y^2+1}}}-{{2}\over{\sqrt{y^2+ 4}}}\right)\;dy}$ = $\displaystyle 4\,\sqrt{5}-3\,2^{{{3}\over{2}}}$ $\displaystyle \int_{1}^{2}{y\,\sqrt{y^2+4}\;dy}$ = $\displaystyle {{2^{{{9}\over{2}}}}\over{3}}-{{5^{{{3}\over{2}}}}\over{3}}$ $\displaystyle \int_{1}^{2}{y\,\left(\sqrt{y^2+4}-\sqrt{y^2+1}\right)\;dy}$ = $\displaystyle -{{5^{{{3}\over{2}}}-2^{{{9}\over{2}}}}\over{3}}-{{5^{{{3}\over{2}} }-2^{{{3}\over{2}}}}\over{3}}$ $\displaystyle \int_{1}^{2}{{{y}\over{\sqrt{y^2+4}}}\;dy}$ = $\displaystyle 2^{{{3}\over{2}}}-\sqrt{5}$

  8. Evaluate $\displaystyle \int_{1}^{2}{\int_{1}^{4}{x^{{{3}\over{2}}}\,\left(y-x\right)\;dx} \;dy}$.

    $\displaystyle \int_{1}^{2}{3\,\left(y-4\right)-8\;dy}$ = $\displaystyle -{{31}\over{2}}$ $\displaystyle \int_{1}^{2}{-y+8\,\left(y-4\right)+1\;dy}$ = $\displaystyle -{{31}\over{2}}$ $\displaystyle {{5\,\int_{1}^{2}{{{320\,y-1024}\over{5}}-{{5\,y-4}\over{20}}\;dy} }\over{2}}$ = $\displaystyle -{{619}\over{35}}$ $\displaystyle \int_{1}^{2}{{{448\,y-1280}\over{35}}-{{14\,y-10}\over{35}}\;dy}$ = $\displaystyle -{{619}\over{35}}$

  9. Evaluate $\displaystyle\int\!\!\int_D x-y dxdy$ where $D$ is the region bounded by $\displaystyle y=2\,x^2$ and $\displaystyle y=x^2+4$.

    $\displaystyle \int_{-3}^{3}{-3\,x^4-x^3+8\,x^2+4\,x+16\;dx}$ = $\displaystyle -{{258}\over{5}}$ $\displaystyle \int_{-2}^{2}{-3\,x^4-x^3+8\,x^2+4\,x+16\;dx}$ = $\displaystyle {{1024}\over{15}}$ $\displaystyle \int_{-2}^{2}{-{{x^4-2\,x^3+8\,x^2-8\,x+16}\over{2}}+2\,x^4-2\,x^3 \;dx}$ = $\displaystyle -{{512}\over{15}}$ $\displaystyle \int_{-3}^{3}{-{{x^4-2\,x^3+8\,x^2-8\,x+16}\over{2}}+2\,x^4-2\,x^3 \;dx}$ = $\displaystyle {{129}\over{5}}$

  10. Evaluate $\displaystyle \int_{0}^{1}{\int_{0}^{1}{e^{y+x}\;dy}\;dx}$.

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


Department of Mathematics
Last modified: 2025-09-14