Math 1210-001 Lab 11
Spring 2016; due Wednesday April 27 by 5:00 p.m. LCB 204
Name _____________________________________________UID______________________________
This lab contains section 5.3, 5.4, 5.6 material. There is a parallel WebWork assignment that gives you
more opportunity for practice. Some problems that you set up in this lab are associated to WebWork
problems where you enter final numerical answers.
1) (See WebWork #1) Consider the region in the first quadrant above the parabola y = 2 x2 and below the
parabola y = 3 K x2 .
1a) Sketch the region and label the point in the first quadrant where the curves intersect.
1b) Consider the volume generated by rotating this region about the y K axis. Find the volume. (You
will want to use cylindrical shells.)
For the next two problems set up the volume integrals for the specified volumes of revolution. It's up to
you to decide whether you're going to use cylindrical shells, or slabs (disks or washers). You don't need
to evaluate the integrals. In all of these problems it is the same region being rotated as in parts a,b on the
previous page.
1c) First quadrant region bounded above by y = 3 K x2 , below by y = 2 x2 and on the left by the y K axis.
Rotate this region about the line y =K1.
1d) First quadrant region bounded above by y = 3 K x2 , below by y = 2 x2 and on the left by the y K axis.
Rotate this region about the line x = 3.
2) (See WebWork #2) The region in the first quadrant above the x K axis and below
y = 1 C sin x , 0 % x % p is rotated about the y K axis. Find the resulting volume. Hint: x$sin x dx
=Kx$cos x C sin x C C.
3a) Use Calculus to find the length of the curve
x = 5 sin t C 1
y = 5 cos t K 6
0%t%p
3b) Identify what geometric object the curve in 3a traces out. (This identification will give you a way to
check your answer to part a.)
4) (See WebWork #5) Find the length of the curve defined by
y=2x
from x = 0 to x = 3.
3
2
5) Use the Calculus formula for surface area of surfaces of revolution in section 5.4 to verify that the
surface area of a sphere of radius R is given by
A = 4 p R2 .
Hint: rotate the graph y =
R2 K x2 about the x K axis.
6) Find the surface area of the conical surface generated by rotating the line y =
x K axis.
1
x, 0 % x % 6 about the
2
7) (See WebWork #9) The density d x a wire at a point x cm from the left end is given by
The wire is 3 cm long.
7a) What is the total mass of the wire?
7b) What is the center of mass of the wire?
1 2 gm
x
.
9
cm
8) (See WebWork #11) Find the centroid of the region in the first quadrant bounded below by the
x K axis and above by y = sin x , for 0 % x % p.
9) Consider the doughnut shaped solid created by rotating the disk of radius 1, centered at 2, 0 about
the y K axis. (Note: The circle bounding this disk has equation
x K 2 2 C y2 = 1.
Set up the volume integral using cylindrical shells. Use Wolfram alpha to evaluate the definite integral
(because the integrand is not one we've learned how to antidifferentiate at this point). Your final answer
2
should have a factor of p . We will discuss "Pappus' Theorem" in class - it gives a shortcut answer to the
volume in terms of the centroid of the disk.