s 1. Find the center of mass of the following plane region with variable density. Describe the distribution of mass in the region: The upper half (y 0) of the y 2 2 disk bounded by the circle x + y =4 with ( x , y )=1+ 2 2. Find the center of mass of the following solids, assuming a constant density of 1. Sketch the region and indicate the location of the centroid. Use symmetry when possible and choose a convenient coordinate system 2 2 a. The sliced solid cylinder bounded by x + y =1, z=0, y + z=1 b. The solid bounded by the upper half (z 0) of the ellipsoid 2 2 2 4 x +4 y + z =16 3. Find the coordinates of the center of mass of the following solid with variable density: The solid bounded by the upper half of the sphere =6z=0 withdensity f ( , , )=1+ 4 4. Determine whether the following statements are true and give an explanation or counterexample: a. A thin plate of constant density that is symmetric about the x-axis has a center of mass with an x-coordinate of zero. b. A thin plate of constant density that is symmetric about both the x-axis and the y-axis has its center of mass at the origin c. The center of mass of a thin plate must lie on the plate d. The center of mass of a connected solid region (all in one piece) must lie within the region 5. Find the mass and center of mass of the thin constant-density plate shown in the figure: 6. Sketch the following vector field: F= x , yx 7. Match vector fields a-d with graphs A-D 8. Determine the points (if any) on the curve C at which the vector field F is tangent to C and normal to C. Sketch C and a few representative vectors of F: F= x , y , whereC= {( x , y ) : x 2 + y 2=4 } 9. Find the gradient field F= for the following potential functions . a. ( x , y )= xy b. ( x , y , z )=ln ( 1+x 2+ y 2+z 2 ) c. 2 2 1 2 2 ( x , y , z ) =( x + y + z ) 10.Scalar line integrals in the plane: a) Find a parametric description for C in the form r ( t ) = x ( t ) , y (t ) . b) Evaluate |r ' (t )| . c) Convert the line integral into an ordinary integral with respect to the parameter and evaluate it x ds ; C isthe line segment 2 x +y 2 a. (1,1 ) (10,10) C b. xy ds ; C is the portion of the ellipse C x2 y2 + =1 in the first quadrant, 4 16 ( )( ) oriented counterclockwise 11.Find the average value of the following function on the given curve: f ( x , y ) = 4+ 9 y 2 3 on the curve 3 y=x 2 , for 0 x 5 12.Use a scalar line integral to find the length of the following curve: r ( t ) = 30 sin ( t ) , 40 sin ( t ) ,50 cos (t ) , for 0 t 2 13.Given the following vector field and oriented curve, C, evaluate F= x, y x2+ y2 F T ds C . on the line r (t )= t , 4 t , for 1 t 10 14.Given the force field F, find the work required to move an object on the given oriented curve. F= y , x on the parabola y =2 x 2 ( 0,0 ) ( 2,8 ) 15.Consider the following vector fields F and closed oriented curves C in the plane (see figures). a) Based on the picture, make a conjecture about whether the circulation of F on C is positive, negative, or zero. b) Compute the circulation and interpret the result a. F= y x , x ; C :r ( t ) = 2 cos ( t ) ,2 sin ( t ) , for 0 t 2 b. F= x , y 2 1 2 2 (x + y ) ; C is the boundary of the square with vertices (2 , 2), traversed counterclockwise