2025年重積分測試題及詳細(xì)答案

一、單項(xiàng)選擇題1.設(shè)\(D\)是由\(x=0\)，\(x=1\)，\(y=0\)，\(y=1\)所圍成的正方形區(qū)域，則\(\iint_Dxydxdy\)的值為（）A.\(\frac{1}{4}\)B.\(\frac{1}{2}\)C.\(\frac{1}{8}\)D.\(\frac{1}{16}\)答案：C2.設(shè)\(f(x,y)\)在有界閉區(qū)域\(D\)上連續(xù)，則\(\iint_Df(x,y)dxdy\)（）A.一定存在B.一定不存在C.可能存在也可能不存在D.以上都不對答案：A3.交換積分次序\(\int_0^1dx\int_x^1f(x,y)dy\)后為（）A.\(\int_0^1dy\int_y^1f(x,y)dx\)B.\(\int_0^1dy\int_0^yf(x,y)dx\)C.\(\int_1^0dy\int_y^1f(x,y)dx\)D.\(\int_1^0dy\int_0^yf(x,y)dx\)答案：A4.設(shè)\(D\)是由\(x^2+y^2\leq1\)所確定的圓域，則\(\iint_De^{-(x^2+y^2)}dxdy\)用極坐標(biāo)計(jì)算時(shí)為（）A.\(\int_0^{2\pi}d\theta\int_0^1e^{-r^2}rdr\)B.\(\int_0^{2\pi}d\theta\int_0^1e^{-r^2}dr\)C.\(\int_0^{\pi}d\theta\int_0^1e^{-r^2}rdr\)D.\(\int_0^{\pi}d\theta\int_0^1e^{-r^2}dr\)答案：A5.已知\(f(x,y)\)為連續(xù)函數(shù)，\(\iint_Df(x,y)dxdy=0\)，其中\(zhòng)(D\)關(guān)于\(x\)軸對稱，則（）A.\(f(x,y)\)是奇函數(shù)B.\(f(x,y)\)是偶函數(shù)C.\(f(x,y)\)可能是奇函數(shù)也可能是偶函數(shù)D.\(f(x,y)\)關(guān)于\(x\)軸的積分值為0答案：D6.設(shè)\(D\)是由\(y=x\)，\(y=2x\)，\(x=1\)所圍成的區(qū)域，則\(\iint_Dxdxdy\)的值為（）A.\(\frac{1}{3}\)B.\(\frac{1}{4}\)C.\(\frac{1}{6}\)D.\(\frac{1}{8}\)答案：C7.若\(f(x,y)\)在區(qū)域\(D\)上可積，且\(D=D_1+D_2\)，則\(\iint_Df(x,y)dxdy\)（）A.\(\iint_{D_1}f(x,y)dxdy+\iint_{D_2}f(x,y)dxdy\)B.\(\iint_{D_1}f(x,y)dxdy-\iint_{D_2}f(x,y)dxdy\)C.\(\iint_{D_1}f(x,y)dxdy\times\iint_{D_2}f(x,y)dxdy\)D.以上都不對答案：A8.設(shè)\(D\)是由\(x\geq0\)，\(y\geq0\)，\(x+y\leq1\)所圍成的區(qū)域，則\(\iint_D(x+y)dxdy\)的值為（）A.\(\frac{1}{3}\)B.\(\frac{1}{4}\)C.\(\frac{1}{6}\)D.\(\frac{1}{8}\)答案：C9.交換積分次序\(\int_0^2dx\int_x^2f(x,y)dy\)后為（）A.\(\int_0^2dy\int_0^yf(x,y)dx\)B.\(\int_0^2dy\int_y^2f(x,y)dx\)C.\(\int_2^0dy\int_0^yf(x,y)dx\)D.\(\int_2^0dy\int_y^2f(x,y)dx\)答案：A10.設(shè)\(D\)是由\(x^2+y^2\leq4\)所確定的圓域，則\(\iint_D\sqrt{4-x^2-y^2}dxdy\)用極坐標(biāo)計(jì)算時(shí)為（）A.\(\int_0^{2\pi}d\theta\int_0^2\sqrt{4-r^2}rdr\)B.\(\int_0^{2\pi}d\theta\int_0^2\sqrt{4-r^2}dr\)C.\(\int_0^{\pi}d\theta\int_0^2\sqrt{4-r^2}rdr\)D.\(\int_0^{\pi}d\theta\int_0^2\sqrt{4-r^2}dr\)答案：A二、多項(xiàng)選擇題1.下列關(guān)于重積分的說法正確的是（）A.重積分的值與積分區(qū)域的劃分方式有關(guān)B.重積分的值與積分變量的選取有關(guān)C.若\(f(x,y)\)在區(qū)域\(D\)上連續(xù)，則\(\iint_Df(x,y)dxdy\)存在D.重積分可化為累次積分進(jìn)行計(jì)算答案：CD2.設(shè)\(D\)是由\(x=0\)，\(x=1\)，\(y=0\)，\(y=1\)所圍成的正方形區(qū)域，\(f(x,y)\)在\(D\)上連續(xù)，則下列積分正確的是（）A.\(\iint_Df(x,y)dxdy=\int_0^1dx\int_0^1f(x,y)dy\)B.\(\iint_Df(x,y)dxdy=\int_0^1dy\int_0^1f(x,y)dx\)C.\(\iint_Df(x,y)dxdy=\int_0^1dx\int_x^1f(x,y)dy+\int_0^1dy\int_y^1f(x,y)dx\)D.\(\iint_Df(x,y)dxdy=\int_0^1dx\int_0^xf(x,y)dy+\int_0^1dy\int_0^yf(x,y)dx\)答案：AB3.設(shè)\(D\)是由\(x^2+y^2\leq1\)所確定的圓域，\(f(x,y)\)在\(D\)上連續(xù)，則\(\iint_Df(x,y)dxdy\)用極坐標(biāo)表示為（）A.\(\int_0^{2\pi}d\theta\int_0^1f(r\cos\theta,r\sin\theta)rdr\)B.\(\int_0^{2\pi}d\theta\int_0^1f(r\cos\theta,r\sin\theta)dr\)C.\(\int_0^{\pi}d\theta\int_0^1f(r\cos\theta,r\sin\theta)rdr\)D.\(\int_0^{\pi}d\theta\int_0^1f(r\cos\theta,r\sin\theta)dr\)答案：A4.若\(f(x,y)\)在區(qū)域\(D\)上可積，且\(D\)關(guān)于\(y\)軸對稱，則（）A.當(dāng)\(f(x,y)\)關(guān)于\(x\)為奇函數(shù)時(shí)，\(\iint_Df(x,y)dxdy=0\)B.當(dāng)\(f(x,y)\)關(guān)于\(x\)為偶函數(shù)時(shí)，\(\iint_Df(x,y)dxdy=2\iint_{D_右}f(x,y)dxdy\)，其中\(zhòng)(D_右\)是\(D\)在\(y\)軸右側(cè)的部分C.\(\iint_Df(x,y)dxdy\)的值與\(f(x,y)\)關(guān)于\(y\)的奇偶性無關(guān)D.當(dāng)\(f(x,y)\)關(guān)于\(y\)為奇函數(shù)時(shí)，\(\iint_Df(x,y)dxdy=0\)答案：ABD5.設(shè)\(D\)是由\(y=x\)，\(y=-x\)，\(x=1\)所圍成的區(qū)域，則\(\iint_Dx^2dxdy\)的值為（）A.\(\int_{-1}^1dx\int_{-x}^xx^2dy\)B.\(\int_{-1}^1dx\int_{x}^{-x}x^2dy\)C.\(\frac{2}{3}\)D.\(\frac{1}{3}\)答案：AC6.交換積分次序后正確的是（）A.\(\int_0^1dx\int_x^1f(x,y)dy=\int_0^1dy\int_0^yf(x,y)dx\)B.\(\int_0^2dx\int_0^{x^2}f(x,y)dy=\int_0^4dy\int_{\sqrt{y}}^2f(x,y)dx\)C.\(\int_0^1dx\int_0^{1-x}f(x,y)dy=\int_0^1dy\int_0^{1-y}f(x,y)dx\)D.\(\int_0^1dx\int_x^1f(x,y)dy=\int_0^1dy\int_y^1f(x,y)dx\)答案：BCD7.設(shè)\(D\)是由\(x^2+y^2\leq9\)所確定的圓域，\(f(x,y)\)在\(D\)上連續(xù)，則\(\iint_Df(x,y)dxdy\)用極坐標(biāo)計(jì)算時(shí)（）A.積分區(qū)域\(D\)：\(0\leqr\leq3\)，\(0\leq\theta\leq2\pi\)B.\(\iint_Df(x,y)dxdy=\int_0^{2\pi}d\theta\int_0^3f(r\cos\theta,r\sin\theta)rdr\)C.不能用極坐標(biāo)計(jì)算D.與直角坐標(biāo)計(jì)算結(jié)果不同答案：AB8.若\(f(x,y)\)在區(qū)域\(D\)上可積，且\(D\)是矩形區(qū)域\(a\leqx\leqb\)，\(c\leqy\leqd\)，則\(\iint_Df(x,y)dxdy\)（）A.\(\int_a^bdx\int_c^df(x,y)dy\)B.\(\int_c^ddy\int_a^bf(x,y)dx\)C.可先對\(x\)積分再對\(y\)積分，也可先對\(y\)積分再對\(x\)積分D.積分結(jié)果與積分次序無關(guān)答案：ABC9.設(shè)\(D\)是由\(y=\sinx\)，\(x=0\)，\(x=\pi\)，\(y=0\)所圍成的區(qū)域，則\(\iint_Dxdxdy\)的值為（）A.\(\int_0^{\pi}dx\int_0^{\sinx}xdy\)B.\(\int_0^{\pi}x\sinxdx\)C.\(\pi\)D.\(\pi^2\)答案：AB10.下列關(guān)于重積分性質(zhì)的說法正確的是（）A.若\(f(x,y)\leqg(x,y)\)在區(qū)域\(D\)上成立，則\(\iint_Df(x,y)dxdy\leq\iint_Dg(x,y)dxdy\)B.\(\iint_Dkf(x,y)dxdy=k\iint_Df(x,y)dxdy\)，\(k\)為常數(shù)C.\(\iint_D[f(x,y)+g(x,y)]dxdy=\iint_Df(x,y)dxdy+\iint_Dg(x,y)dxdy\)D.重積分的值只與被積函數(shù)和積分區(qū)域有關(guān)答案：ABCD三、判斷題1.重積分的值與積分區(qū)域的形狀有關(guān)。（）答案：對2.在計(jì)算重積分時(shí)，積分變量的選取不影響積分結(jié)果。（）答案：對3.若\(f(x,y)\)在區(qū)域\(D\)上有界，則\(\iint_Df(x,y)dxdy\)一定存在。（）答案：錯(cuò)4.交換積分次序時(shí)，積分上下限的變化是根據(jù)積分區(qū)域的特點(diǎn)來確定的。（）答案：對5.當(dāng)積分區(qū)域關(guān)于\(x\)軸對稱且\(f(x,y)\)關(guān)于\(y\)為奇函數(shù)時(shí)，\(\iint_Df(x,y)dxdy=0\)。（）答案：對6.用極坐標(biāo)計(jì)算重積分時(shí)，\(dxdy=rdrd\theta\)。（）答案：對7.重積分可用于計(jì)算平面圖形的面積。（）答案：對8.若\(f(x,y)\)在區(qū)域\(D\)上連續(xù)，則\(\iint_Df(x,y)dxdy\)可以通過將\(D\)劃分成小區(qū)域，然后取極限得到。（）答案：對9.對于重積分\(\iint_Df(x,y)dxdy\)，如果\(D\)是圓形區(qū)域，用極坐標(biāo)計(jì)算一定比直角坐標(biāo)計(jì)算簡單。（）答案：錯(cuò)10.若\(f(x,y)\)在區(qū)域\(D\)上可積，且\(D=D_1-D_2\)，則\(\iint_Df(x,y)dxdy=\iint_{D_1}f(x,y)dxdy-\iint_{D_2}f(x,y)dxdy\)。（）答案：對四、簡答題1.簡述重積分的概念。重積分是定積分在多元函數(shù)中的推廣。對于二元函數(shù)\(f(x,y)\)在有界閉區(qū)域\(D\)上的重積分，是通過將區(qū)域\(D\