Parameterized Curves 1. Find a parameterization for the curve shown:
2. 3. 4. 5. 6.
Find the line in the direction of the vector and through the point (-3, 4, -2). Find the line through (3, -2, 2) and intersecting the y-axis at y=2. The circle of
17.3 -17.4 Vector Fields and Flow
1. Find formulas for the vector fields below:
2. Sketch the vector fields: a. b. c. Challenge Problem: a) Let . Find a point at which is parallel to l, the line x = 5+t, y=6-3t, z=7-3t b) Find a point at which they are pe
18. 1 Line Integrals 1. Say whether you expect the line integral of the pictured vector fields over the given curve to be positive, negative or zero.
from (-1, 0) to (1, 0). Do the line integrals of the vector field along each of the paths C1, C2 and C3 a
Solutions to Recitation Problems Chapter 18 18.1
5. 18.2 1a.
2a-d) a)Path independent vector fields are constant b)Not path independent because the line integral around a closed curve around the origin in
19.1 Flux Integrals 1. Find the flux of the constant vector field through the surface given by the triangular plate of area 4 in the yz-plane oriented in the positive x-direction. 2. Compute the flux of the vector field through the rectangular region show
Applications of Integration to Probability 1. Let p be the joint density function such that p(x,y) = xy in R, the rectangle
. b. c. d. Within a distance 1 from the origin e. 2. Check whether p is a joint density function. Assume p(x,y) =0 outside the regi
Volume Integrals Sheet 2 Volume Integrals in Polar Coordinates 1. For the regions R below, write as an iterated integral in polar coordinates.
a. b. c. 2. Choose rectangular and polar coordinates to set up an iterated integral of an arbitrary function f(
1. Find the triple integral of the function over the region. (, , ) = - , W is the rectangular box with corners (0,0,0), (a,0,0), (0,b,0) and (0,0,c). 2. Find the triple integral of the function over the region. (, , ) = ( + ), W is the cube 0 , 0 , 0 . 3
Volume Integrals Sheet 3 Integrals in Cylindrical and Spherical Coordinates
1. Match the equations (a)-(f) with one of the surfaces in I-VIII.
b. (, , ) = sin( + ), W is the solid cylinder with height 4 and with base of radius 1 and centered on the z axis
Mock- Exam Math 2130 Spring 2012
Problem 2: A point is chosen at random from the region S in the xy-plane containing all points (x,y) such that
a) Determine the joint density function for x and y b) If T is a subset of S with area , then find the probabil
Homework 9 Solutions
Section 17.3 # 20,22,23,32 Section 17.4 # 1,7,8,16,22 p920:59c; p922:2
32. a) The gradient is perpendicular to the level curves. A function always increases in the direction of its gradient; this is why the values of the level curves