Math 12. Calculus Plus. Written Homework 6.
Due on Wednesday, 10/30/13.
Turn in this homework in the Math 12 box in Kemeny Hall before 10 am on Wednesday.
RR
1. (Ch 15.10, #18) Evaluate R (x2 − xy + y 2 ) dA, where R is the region bounded by the ellipse
p
p
√
√
x2 − xy + y 2 = 2. Use the change of variables x = 2 u − 2/3 v, y = 2 u + 2/3 v.
2. (Ch 15.10, #14) Let R be the region bounded by hyperbolas y = 1/x, y = 4/x, and the lines
y = x, y = 4x, in the first quadrant. Find equations for a transformation T that maps a
rectangular region S in the uv-plane onto R, where the sides of S are parallel to the u- and
v- axes.
RR
3. (Ch 15.10, #19) Use the transformation x = u/v, y = v to evaluate the integral R xy dA,
where R is the region in the first quadrant bounded by the lines y = x and y = 3x and the
hyperbolas xy = 1, xy = 3.
4. (Ch 15.10, #23) Use an appropriate change of variables to evaluate
ZZ
x − 2y
dA,
R 3x − y
where R is the parallelogram enclosed by the lines x − 2y = 0, x − 2y = 4, 3x − y = 1 and
3x − y = 8.
5. (Ch 16.2, #16) Evaluate the line integral
Z
(y + z)dx + (x + z)dy + (x + y)dz,
C
where C is the concatenation of the line segment from (0, 0, 0) to (1, 0, 1) with the line segment
from (1, 0, 1) to (0, 1, 2).
6. (Ch 16.2, #34) A thin wire has the shape of the first-quadrant part of the circle with center
the origin and radius a. If the density function is ρ(x, y) = kxy, find the mass and center of
mass of the wire.
p
7. (Ch 16.1, #26) Find the gradient vector field ∇f of f (x, y) = x2 + y 2 and sketch it.
8. (Ch 16.2, #42) The force exerted by a unit electric charge at the origin on a charged particle
at a point (x, y, z) is F(r) = r/|r|3 , where r = hx, y, zi. Find the work done as the particle
moves along a straight line from (2, 0, 0) to (2, 1, 5).