Math 1210-001 Lab 10
Spring 2016; due April 21 at the beginning of lab
Name _____________________________________________UID______________________________
This lab contains section 4.5, 5.1, 5.2 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) (10 points)
1a) Suppose the temperature in degrees Fahrenheit of a metal bar of length 4 feet along the interval
K2 % x % 2 feet is given according to the formula
T x = 30 C 9 x 4 K x2 .
Find the average temperature of the bar. Use properties of even and odd functions to simplify your
computation.
1b) (See WebWork #1) What is the average temperature of the right half of the metal bar, i.e. the part
along the interval 0 % x % 2 ?
2) (10 points) Use properties of even and odd functions to simplify the following integral computation
p
x3 C 4 cos
Kp
x
2
C 3 sin 2 x cos 2 x dx.
3) (30 points) Sketch the regions bounded by the graphs of the given equations and then compute the area
between the curves (using integrals).
3a) y = x2 K x K 6, y = 0.
3b) x = 4 y K y2 , x = 0.
3c) y = x2 K 2 x, y =Kx2
3d) (WebWork #6) y2 = 4 x, x K
y
K 2 = 0.
2
4) (10 points) An object moves along a line so that its velocity at time t seconds is v t = t2 K 12 t C 60
feet
. Find (a) the displacement, and (b) the total distance traveled by the object, for 0 % t % 6. (Hint:
sec
One of these corresponds to a "signed area" and the other is the "total area" between the velocity graph and
the tKaxis. You've solved similar problems before, by explicitly finding and using the position function
s t . Because of the FTC these problems can alternately be worked using definite integrals.)
5) (40 points) Sketch the regions bounded by the specified equation graphs below. Then set up the
volume integrals for the solids generated by revolving the regions around the specified axes. Note: In
these problems you are only being asked to set up the integrals, not evaluate them.
5a) Region bounded by y =
x , y = 0, x = 4; rotate the region about the x K axis.
5b) (Same) region bounded by y =
x , y = 0, x = 4; rotate the region about the y K axis.
5c) (Same) region bounded by y =
x , y = 0, x = 4; rotate the region about the line y =K1.
5d) (WebWork #11) Region in the first quadrant bounded by y = x2 , y = 3 x; rotate the region about the
x K axis.
5e) (Same) region bounded by y = x2 , y = 3 x; rotate the region about the y K axis.
5f) (Same) region bounded by y = x2 , y = 3 x; rotate the region about the line x = 4.
5g) (Special case of WebWork #9) Region bounded below by y = 0 and above by x2 C y2 = 9. Rotate
this half-disk about the x K axis (to generate a solid ball of radius 3).
5h) (WebWork #9) Region bounded below by y = 0 and above by the upper half of the ellipse
x2
y2
C
= 1. Rotate this region about the x K axis (to generate an ellipsoid).
a2
b2