Instructions: Your homework is in two parts, which you should turn in separately. This is not the same as the week 8 course packet homework. There are too many differences to list them all here; just print out this and use it instead of the course packet homework. Due Wednesday, November 22, in lecture. Week 8 Homework Problems, part A 1 2 Stewart, section 8.1: #1, 2, 9, 15, 23 (omit calculator part), 30 Stewart, section 8.3: #23, 25, 29, 32
Week 8 Homework Problems, part B 3 Let k be greater than 1.
a) Write a definite integral for the arclength of y = xk from x = 0 to x = b. Do not try to solve the integral.
3 b) One case when this integral can be easily evaluated is when k = 2 . In that case use a substitution to evaluate the integral and find a formula for the arclength in terms of b.
c) Use an inverse trig substitution to find a formula for the arclength in the case when k = 2. d) Use Simpson's Rule with 6 sub-intervals to estimate the arclength in the case when k = 3 and b = 1.
4 Consider a uniform flat plate bounded by the graph of y = 1/(1 + x2 ), the graph of y = -1/(1 + x2 ) and the y-axis. Show that the plate has finite mass, but does not have center of mass at a finite distance. (This means that you could lift up the plate, but you could not balance it!)
5 Find the x-coordinate of the center of mass of the uniform flat plate bounded by the x- and y-axes, the line x = 2 and the curve 1 y= . x2 + 6x + 13
y
2 x
6
The formula for the arc length of a curve given parametrically by (x(t), y(t)), for a t b, is
b
L=
a
(x (t))2 + (y (t))2 dt.
A path of a point on the edge of a roll...