課程名稱︰ 微積分3
課程性質︰ 物理系必帶
課程教師︰ 李瑩英
開課學院： 理學院
開課系所︰ 數學系
考試日期（年月日）︰ 2019/04/11
考試時限（分鐘）： 120
試題 :
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.