课程名称︰ 微积分3 课程性质︰ 物理系必带 课程教师︰ 李莹英 开课学院： 理学院 开课系所︰ 数学系 考试日期（年月日）︰ 2019/04/11 考试时限（分钟）： 120 试题 : Calculus 3 Quiz 2 1.(10pts)Evaluate the integral 2 1 ∫∫y*cos(x^3-1)dxdy. 0 y/2 2.(10pts)Evaluate ∞ I=∫e^(-x^2)dx.(hint: consider I^2 and use the double integral) -∞ 3.(10pts)Find the surface area for the part of the sphere x^2+y^2+z^2=1 that lies within the cylinder x^2+y^2 = ax and above the xy-plane. 4.(10pts)Evaluate the triple integral ∫∫∫6xy dV, where E lies under the plane z = 1+x+y and above the region in the xy-plane bounded by the curve y = x^0.5, y = 0, and y = 1. E 5.(10pts)Evaluate ∫∫ y^2 dA, wherre R is the region bounded by the curves xy = 1, xy = 2, xy^2 = 1, xy^2 = 2. 6.(10pts)Find the equations of the normal plane and osculating plane of the curve x = lnt, y = 2t, z = t^2 at (0,2,1). 7.(10pts)The region E lies between the paraboloid z = 24-x^2-y^2 and the cone z = 2(x^2+y^2)^0.5. Find the centroid (the center of mass when the density is constant) of E. 8.(10pts)Find the volume of the solid that lies above the cone z = (x^2+y^2)^0.5 and below the sphere x^2+y^2+z^2 = z. 9.Suppose that the region E = {(x,y,z)|x^2/a^2+y^2/b^2+z^2/c^2 <= 1} has constant density 1: a.(10pts) Compute the moment of the inertia about the z-axis of E. b.(10pts) Let C be the boundary curve of the intersection of E and xy-plane. Compute the curvature of C by finding a parametrization for the curve. --

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