2重積分考試題及答案

一、單項選擇題（每題2分，共10題）1.設(shè)\(D=\{(x,y)|x^{2}+y^{2}\leq1\}\)，則\(\iint_{D}dxdy=\)（）A.\(\pi\)B.\(2\pi\)C.\(1\)D.\(0\)答案：A2.若\(D=\{(x,y)|0\leqx\leq1,0\leqy\leq1\}\)，\(f(x,y)=x+y\)，則\(\iint_{D}f(x,y)dxdy=\)（）A.\(\frac{1}{2}\)B.\(1\)C.\(\frac{3}{2}\)D.\(2\)答案：B3.交換積分次序后\(\int_{0}^{1}dx\int_{x}^{1}f(x,y)dy=\)（）A.\(\int_{0}^{1}dy\int_{0}^{y}f(x,y)dx\)B.\(\int_{0}^{1}dy\int_{y}^{1}f(x,y)dx\)C.\(\int_{0}^{1}dy\int_{0}^{1}f(x,y)dx\)D.\(\int_{1}^{0}dy\int_{0}^{y}f(x,y)dx\)答案：A4.設(shè)\(D=\{(x,y)|x^{2}+y^{2}\leq4\}\)，\(I=\iint_{D}(x^{2}+y^{2})dxdy\)，采用極坐標(biāo)變換\(x=r\cos\theta,y=r\sin\theta\)，則\(I=\)（）A.\(\int_{0}^{2\pi}d\theta\int_{0}^{2}r^{3}dr\)B.\(\int_{0}^{2\pi}d\theta\int_{0}^{2}r^{2}dr\)C.\(\int_{0}^{\pi}d\theta\int_{0}^{2}r^{3}dr\)D.\(\int_{0}^{\pi}d\theta\int_{0}^{2}r^{2}dr\)答案：A5.設(shè)\(D=\{(x,y)|0\leqx\leq1,0\leqy\leqx\}\)，\(I=\iint_{D}e^{x+y}dxdy\)，則\(I=\)（）A.\(e-1\)B.\(e^{2}-1\)C.\(\frac{1}{2}(e-1)^{2}\)D.\(\frac{1}{2}(e^{2}-1)\)答案：C6.設(shè)\(D\)是由\(y=x,y=0,x=1\)所圍成的閉區(qū)域，則\(\iint_{D}xydxdy=\)（）A.\(\frac{1}{8}\)B.\(\frac{1}{4}\)C.\(\frac{1}{2}\)D.\(1\)答案：A7.設(shè)\(I=\iint_{D}\frac{1}{(x^{2}+y^{2})^{2}}dxdy\)，其中\(zhòng)(D=\{(x,y)|1\leqx^{2}+y^{2}\leq4\}\)，采用極坐標(biāo)變換后\(I=\)（）A.\(\int_{0}^{2\pi}d\theta\int_{1}^{2}\frac{1}{r^{3}}dr\)B.\(\int_{0}^{2\pi}d\theta\int_{1}^{2}\frac{1}{r^{2}}dr\)C.\(\int_{0}^{\pi}d\theta\int_{1}^{2}\frac{1}{r^{3}}dr\)D.\(\int_{0}^{\pi}d\theta\int_{1}^{2}\frac{1}{r^{2}}dr\)答案：A8.設(shè)\(D=\{(x,y)|-1\leqx\leq1,-1\leqy\leq1\}\)，\(I=\iint_{D}(x^{2}+y^{2})dxdy\)，則\(I=\)（）A.\(\frac{4}{3}\)B.\(\frac{8}{3}\)C.\(\frac{2}{3}\)D.\(\frac{1}{3}\)答案：A9.設(shè)\(D=\{(x,y)|0\leqy\leq\sqrt{1-x^{2}}\}\)，則\(\iint_{D}dxdy=\)（）A.\(\frac{\pi}{4}\)B.\(\frac{\pi}{2}\)C.\(\pi\)D.\(2\pi\)答案：A10.設(shè)\(D=\{(x,y)|x^{2}+y^{2}\leqR^{2}\}\)，\(I=\iint_{D}x^{2}dxdy\)，則\(I=\)（）A.\(\frac{\piR^{4}}{4}\)B.\(\frac{\piR^{2}}{2}\)C.\(\frac{\piR^{4}}{2}\)D.\(\piR^{4}\)答案：A二、多項選擇題（每題2分，共10題）1.設(shè)\(D=\{(x,y)|x^{2}+y^{2}\leq1\}\)，對于\(\iint_{D}f(x,y)dxdy\)，下列變換正確的是（）A.令\(x=r\cos\theta,y=r\sin\theta\)，\(r\in[0,1],\theta\in[0,2\pi]\)B.令\(x=u,y=v\)，\(u^{2}+v^{2}\leq1\)C.令\(x=\frac{u}{\sqrt{u^{2}+v^{2}}},y=\frac{v}{\sqrt{u^{2}+v^{2}}}\)，\(u^{2}+v^{2}\leq1\)D.令\(x=\cosu,y=\sinu\)，\(u\in[0,2\pi]\)答案：AB2.設(shè)\(D=\{(x,y)|0\leqx\leq1,0\leqy\leq1\}\)，則\(\iint_{D}(x+y)dxdy\)等于（）A.\(\iint_{D}xdxdy+\iint_{D}ydxdy\)B.\(\int_{0}^{1}dx\int_{0}^{1}(x+y)dy\)C.\(\int_{0}^{1}dy\int_{0}^{1}(x+y)dx\)D.\(\frac{1}{2}+\frac{1}{2}\)答案：ABCD3.設(shè)\(D\)是由\(y=x^{2}\)與\(y=x\)所圍成的區(qū)域，則\(\iint_{D}f(x,y)dxdy\)可化為（）A.\(\int_{0}^{1}dx\int_{x^{2}}^{x}f(x,y)dy\)B.\(\int_{0}^{1}dy\int_{y}^{\sqrt{y}}f(x,y)dx\)C.\(\int_{0}^{1}dx\int_{x}^{\sqrt{x}}f(x,y)dy\)D.\(\int_{0}^{1}dy\int_{y^{2}}^{y}f(x,y)dx\)答案：AB4.對于\(\iint_{D}e^{x^{2}+y^{2}}dxdy\)，其中\(zhòng)(D=\{(x,y)|x^{2}+y^{2}\leq1\}\)，下列說法正確的是（）A.用極坐標(biāo)變換\(x=r\cos\theta,y=r\sin\theta\)后計算方便B.其值大于\(e\)C.其值小于\(e\)D.直接用直角坐標(biāo)很難計算答案：ACD5.設(shè)\(D=\{(x,y)|0\leqx\leq1,0\leqy\leq1\}\)，\(I=\iint_{D}\sin(x+y)dxdy\)，則（）A.\(I=\int_{0}^{1}dx\int_{0}^{1}\sin(x+y)dy\)B.\(I=\int_{0}^{1}dy\int_{0}^{1}\sin(x+y)dx\)C.\(I=2\sin1\)D.\(I=\sin1-\cos1+1\)答案：AB6.設(shè)\(D=\{(x,y)|-1\leqx\leq1,-1\leqy\leq1\}\)，對于\(\iint_{D}x^{3}y^{3}dxdy\)，有（）A.其值為\(0\)B.關(guān)于\(x\)是奇函數(shù)，關(guān)于\(y\)是奇函數(shù)C.利用區(qū)域的對稱性和函數(shù)的奇偶性可得出結(jié)果D.直接計算結(jié)果為\(0\)答案：ABCD7.設(shè)\(D=\{(x,y)|x^{2}+y^{2}\leq4\}\)，\(I=\iint_{D}\sqrt{x^{2}+y^{2}}dxdy\)，采用極坐標(biāo)變換后（）A.\(I=\int_{0}^{2\pi}d\theta\int_{0}^{2}r^{2}dr\)B.\(I=\int_{0}^{\pi}d\theta\int_{0}^{2}r^{2}dr\)C.\(I=\frac{8\pi}{3}\)D.\(I=\frac{16\pi}{3}\)答案：AC8.設(shè)\(D=\{(x,y)|0\leqx\leqa,0\leqy\leqb\}\)，則\(\iint_{D}xydxdy=\)（）A.\(\frac{1}{4}a^{2}b^{2}\)B.\(\int_{0}^{a}xdx\int_{0}^ydy\)C.\(\frac{1}{2}a^{2}\frac{1}{2}b^{2}\)D.\(\int_{0}^ydy\int_{0}^{a}xdx\)答案：ABCD9.設(shè)\(D=\{(x,y)|y\geqx^{2},y\leq1\}\)，\(\iint_{D}f(x,y)dxdy\)可化為（）A.\(\int_{-1}^{1}dx\int_{x^{2}}^{1}f(x,y)dy\)B.\(\int_{0}^{1}dy\int_{-\sqrt{y}}^{\sqrt{y}}f(x,y)dx\)C.\(\int_{-1}^{0}dx\int_{x^{2}}^{1}f(x,y)dy+\int_{0}^{1}dx\int_{x^{2}}^{1}f(x,y)dy\)D.\(\int_{0}^{1}dy\int_{-\sqrt{y}}^{\sqrt{y}}f(x,y)dx+\int_{-1}^{0}dx\int_{x^{2}}^{1}f(x,y)dy\)答案：ABC10.設(shè)\(D=\{(x,y)|x^{2}+y^{2}\leq9\}\)，\(I=\iint_{D}(x^{2}+y^{2})dxdy\)，采用極坐標(biāo)變換\(x=r\cos\theta,y=r\sin\theta\)，則（）A.\(I=\int_{0}^{2\pi}d\theta\int_{0}^{3}r^{3}dr\)B.\(I=\frac{81\pi}{2}\)C.\(I=\int_{0}^{\pi}d\theta\int_{0}^{3}r^{3}dr\)D.\(I=\frac{243\pi}{4}\)答案：AB三、判斷題（每題2分，共10題）1.設(shè)\(D=\{(x,y)|x^{2}+y^{2}\leq1\}\)，\(\iint_{D}1dxdy=\pi\)。（）答案：正確2.若\(D=\{(x,y)|0\leqx\leq1,0\leqy\leq1\}\)，\(\iint_{D}(x+y)dxdy=1\)。（）答案：正確3.對于\(\iint_{D}f(x,y)dxdy\)，交換積分次序不改變積分值。（）答案：錯誤4.設(shè)\(D=\{(x,y)|x^{2}+y^{2}\leq4\}\)，\(\iint_{D}(x^{2}+y^{2})dxdy\)用極坐標(biāo)變換計算時，\(x=r\cos\theta,y=r\sin\theta\)，\(r\in[0,2],\theta\in[0,2\pi]\)。（）答案：正確5.設(shè)\(D=\{(x,y)|0\leqx\leq1,0\leqy\leqx\}\)，\(\iint_{D}e^{x+y}dxdy=\frac{1}{2}(e-1)\)。（）答案：錯誤6.設(shè)\(D\)是由\(y=x,y=0,x=1\)所圍成的閉區(qū)域，\(\iint_{D}xydxdy=\frac{1}{8}\)。（）答案：正確7.設(shè)\(I=\iint_{D}\frac{1}{(x^{2}+y^{2})^{2}}dxdy\)，其中\(zhòng)(D=\{(x,y)|1\leqx^{2}+y^{2}\leq4\}\)，采用極坐標(biāo)變換后\(I=\int_{0}^{2\pi}d\theta\int_{1}^{2}\frac{1}{r^{3}}dr\)。（）答案：正確8.設(shè)\(D=\{(x,y)|-1\leqx\leq1,-1\leqy\leq1\}\)，\(\iint_{D}(x^{2}+y^{2})dxdy=\frac{8}{3}\)。（）答案：錯誤9.設(shè)\(D=\{(x,y)|0\leqy\leq\sqrt{1-x^{2}}\}\)，\(\iint_{D}dxdy=\frac{\pi}{2}\)。（）答案：錯誤10.設(shè)\(D=\{(x,y)|x^{2}+y^{2}\leqR^{2}\}\)，\(\iint_{D}x^{2}dxdy=\frac{\piR^{4}}{4}\)。（）