2
Correctly reverse the order of integration, then evaluate
1∫0
1∫y
xeydxdy

3
The plane region R is bounded by the graphs of y=x and y=x2 .
Find the volume over R and beneath the graph of f(x, y) = x + y.

4
Find the volume of the "ice cream cone" bounded by the sphere x2+y2+z2=1 and the cone
z
=
^/¯x2+y2-1

5
Find the moment of inertia around the z-axis of the solid bounded by
x = 0, y = 0, z = 0, y = 1 - x2, and 4x + 3y + 2z = 12, assume _(x,y,z)_1.
6
Evaluate ∫CP(x,y)dx+Q(x,y)dy
Given: P(x,y)=y2, Q(x,y)= 3x; C is the part of the graph of y=3x2 from (-1,3) to (2,12).

7
Use the divergence theorem to evaluate
∫∫s F?nds where n is the outer unit normal vector to the surface S. ∫∫F = 3xi + 2y2j + 4zk;
S is the surface of the plane x + y + z = 6.
8
Use Stokes theorem to evaluate W = F?Tds where F(x, y, z) = 3yi - 2xj +4xk; C is the circle: x2 + y2 = 9, z = 4,
oriented counterclockwise as viewed from above.

(See attached file for full problem description with equations)
(Steiner's theorem) If IA is the moment of inertia of a mass distribution of total mass M with respect to an axis A through the center of gravity, show that its moment of inertia IB with respect to an axis B, which is parallel to A and has the distance k from it,

Let F = (2x, 2y, 2x + 2z). Use Stokes' theorem to evaluate the integral of F around the curve consisting of the straight lines joining the points (1,0,1), (0,1,0) and (0,0,1). In particular, compute the unit normal vector and the curl of F as well as the value of the integral:

StokesTheorem. See attached file for full problem description.
1. compute the line integral where F = (yz^2 - y)i + (xz^2 + x)j + 2xyzk where C is the circle of radius 3 in the xy-plane, centered at the origin, oriented counterclockwise as viewed from the positive z -axis.
2. Given F =yi - xj + yzk and the region S determ

Using Green's TheoremandStokes' Theorem respectively, calculate the given line integrals.
• Using Green's Theorem calculate the line integral , where along the positively oriented closed curve C which is the boundary of the domain: .
Which line integrals you would have to evaluate instead in order to calculate h

Given the vector field F=3yi + (5-2x)j + ((z^2)-2)k find
a) Div F
b) Curl F
c) The surface integral of the normal component of curl F over the open hemispherical surface (x^2)+(y^2)+(z^2)=4 above the xy plane.
* Hint: by a double application of Stokestheorem, part c can be reduced to a triviality

7.. Given the vector field F(x,y,z) = xi + (x+2y+3z) j + z2 k
Let C he the circle on the xy-plane, centered at the origin (0,0) and having as radius r=5. Let S be the part of the paraboloid z = 16? x2 ? y2 which lies above the xy-plane (z ≤ 0). Use the Stokes's Theorem to evaluate the line integral of this vector field a

To solve many problems about rotational motion, it is important to know the moment of inertia of each object involved. Calculating the moments of inertia of various objects, even highly symmetrical ones, may be a lengthy and tedious process. While it is important to be able to calculate moments of inertia from the definition, in

An irregular piece of sheet metal mounted on a pivot at a distance of .62 m from the cm. About this pivot point, it oscillates with SHM with a period measured at 2.25 seconds.
Part A: Find the moment of inertia about the pivot axis.
Part B: Find the moment of inertia about the cm axis.