Math 280 Name ______
Fall 2016
Exam 3
No Work = No Credit!
1. Set up an integral for both orders of integration, and use the more convenient order to evaluate the integral over the region R.
Ryx2+y2dA
R: Triangle bounded by y=x,y=2x,x=2
2. Set up a triple integral for the volume of each solid region. You do not need to evaluate the integral.
a) The region in the first octant bounded by the cylinder z=1-y2and lying between the vertical planes x+y=1and x+y=3.
b) The upper hemisphere given by z=1-x2-y2.
c) The region bounded below by the paraboloid z=x2+y2and above by the sphere x2+y2+z2= 6.
3. Let R be the region bounded by the lines
x-2y=0,x-2y=-4,x+y=4,x+y=1
Evaluate the double integral R3xydA by using an appropriate change of variables.
4. Evaluate EzdVwhere E is the solid tetrahedron bounded by the 4 planes
x=0,y=0,z=0,x+y+z=1
5. Evaluate Cydx+x2dy where C is the parabolic arc given by y=4x-x2 from (4, 0) to (1, 3).
6. Use spherical coordinates to find the volume of the solid that lies above the cone z=x2+y2 and below by the sphere x2+y2+z2=z.
7. Use Green’s Theorem to evaluate the line integral.
Csinxcosydx+xy+cosxsinydy
Where C is the boundary of the region lying between the graphs of y = x and y=x
8. Compute the work done by the force field F=y,-xacting on an object as it moves along the parabola y=x2-1from (1, 0) to -2,3.
9. Show that the line integral is independent of path and then use that fact to evaluate the integral.
C2xcosz-x2dx+z-2ydy+y2-x2sinzdz
where C runs from3,-2,0 to 1,0,π.
10. Determine if the vector field is conservative and/or incompressible.
F=2xz,y2+z2,zy2
11. Let f be a scalar field and let F be a vector field. State whether each expression is meaningful. If not explain why. If so, state whether the result will be a vector field or a scalar field.
a) curl f
b) curl (grad f)
c)
d)
e)
f)
g)
12. Find the area of the surface cut from the bottom of the paraboloid by the plane
z = 4.
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