Solutions to Homework 1
1. Let f (x) = x2 , a = 1, b = 2, and let x0 = a = 1, x1 = 1.1, x2 = 1.2, x3 = 1.4, x4 = b = 2. Let P = (x0 , . . . , x4 ), so that P is a partition of the interval [1, 2]. List the following: each of the subintervals, each of the
LB 119 Homework 2 Solutions
1. Use a cross section method (e.g. disk, washer, or shell method your choice) to demonstrate that the volume of a right circular cone of radius r and height h is equal to r2 h. 3 Solution: Consider the figure below.
The region
Solutions to Homework 3
1. Determine the centroid the region in plane which lies above the the x-axis and below the semi-circle y = r2 - x2 . Solution: The centroid is the same as the center of mass where the density has a constant value. The convention i
Solutions to Homework 4
1. The following formulas are easy to remember due to their similarity: 1 sin2 x dx = (x - sin x cos x) + C 2 1 cos2 x dx = (x + sin x cos x) + C. 2 Show that the above formulas are true. (Hint: You may want to use more than one tr
Solutions to Homework 5
1. Consider the series
j=2
3 . 22j
(a) Re-index the series so that it has the form ak . k=1 Solution: Lower the index j by one and replace each appearance of j in the terms of the series by (j + 1):
j=1
3 22(j+1)
=
j=1
3 22j+2
.
A
LB 119 Homework 6 (due Wednesday, 11/14/12)
Directions. Please work on the problems below. Your solutions must begin with a clear statement (or re-statement in your own words) of the problem. You solutions should be clear, legible, and demonstrate, at min
Vectors
Example: The Jet Stream
A jet stream refers to an air current which affects the speed of a jet flying
at a high altitude. Suppose that a certain jet stream increases the speed of
a jet by 30 miles per hour (mph) if the jet ies parallel to the ow