A. (15 pts.) Find the arclength of the path
from t = 1 to t = 3
(Hint: once you have your differential of arc length, factor it.).
B. (20 pts.) Find the curl of the vector field
What does this tell us about whether or not F is a gradient vector field?
C. (20 pts.) Find the line integral of the vector field
F(x,y) = (x2-y2,y3-2xy)
along the path c(t) = (t,1-t) , 0 £ t £ 1.
D. (20 pts.) Find the area of the region D in the plane whose boundary is the parametrized curve
c(t) = (4t-t3,2t-t2), 0 £ t £ 2.
E. Find the velocity and acceleration of the parametric curve
F. Find the volume of the region T lying between the sphere r = 3 (in spherical coordinates) and the cone f = p/6.
G. (20 pts.) Find the integral of the function
over the region lying under the graph of the function z = x2 and over the region in the x-y plane with x2+y2 £ 4 and y ³ 0 .
(Hint: this is probably most easily done dz dy dx).
H. (20 pts.) Find the volume of the region lying under the graph of the function
which lies over the circle of radius 3 in the x-y plane centered at the origin.
J. (20 pts.) A particle moves along a curve C in 3-space, starting at time t = 0 at the point (1,0,1), and at every time t, it's velocity vector is given by
What is the particle's position at time t = 2 ?
(Hint: how do you determine f(t), knowing f¢(t) and f(0) ?)
K. (20 pts.) Show that the vector field
is a conservative vector field, find a potential function for [F\vec], and use this function to compute the line integral of [F\vec] over the parametrized curve
L. (20 pts.) Use Green's theorem to compute the line integral of the vector field
over the curve which follows the line segments from (0,0) to (2,0) to (0,1) to (0,0).
M. (20 pts.) Find the flux integral of the vector field
over that part of the graph of the function
which lies over the triangle in the plane with vertices (0,0),(1,0), and (1,2) (and using the upward pointing normal for the surface).
N. (25 pts.) Use a change of variables to find the integral of the function
over the parallelogram P with vertices (0,0), (1,1), (3,1), and (4,2).
(Hint: Find the (linear!) map f which takes the unit square [0,1]×[0,1] to P.)