Find the area of the parallelogram with vertices

1. (a) Find the area of the parallelogram with verticesA(????2;1);B(0;4);C(4;2);andD(2;????1)

(b) Find the volume of the parallelepiped determined by the vectorsa=<1;2;3>,

b =<????1;1;2>andc=<2;1;4>. [10 points]

2. Find the area of the region inside the circle (x????1)2+y2= 1 and outside the circle x2+y2= 1.

[10 points]

3. Determine whetherL1:x=y=z; L2:x+ 1 =y/2 =z/3 are parallel, skew, or intersecting. If

skew, _nd the distance between them. [12 points]

4. Find the velocity and position vectors of a particle that has the given acceleration and the given initial velocity and positiona(t) = 2i+ 6tj+ 12t2k;v(0) =i;r(0) =j????k. [10 points]

5. Evaluate the line integralZCF_dr, whereF=< x+2y; x2>andCconsists of the line segments from (0;0) to (2;1) and from (2;1) to (3;0). [12 points]

6. A solidElies within the cylinderx2+y2= 1, below the planez= 4 and above the paraboloid

z= 1????x2????y2. The density function is_(x; y) =px2+y2. Find the mass ofE. [10 points]

7. Find the absolute maximum and minimum values off =x2+y2????2xon the setD, whereDis

the closed triangular region with vertices (2;0), (0;2), and (0;????2). [12 points]

8. Evaluate the integral

Z1

0

Zp

1????x2

0

Zp

2????x2????y2

p

x2+y2

xydzdydx

[12 points]

9. Evaluate the surface integralZZSF_dS, whereF=<3xy2; xez; z3>andSis the surface of the solid bounded by the cylindery2+z2= 1 and the planesx=????1 andx= 2. [12 points].

10. (a) Show that the equation of the tangent plane to the ellipsoidx2/a2+y2/b2+z2/c2=R2atP(x0; y0; z0) can be written asxx0/a2+yy0/b2+zz0/c2=R2

(b) Use the result of part (a) to _nd the tangent plane tox2/4 +y2/4 +z2= 1 at the point (????1;1;

P2

2

).

[Extra 10 points]