Math 241, Lab 4 (corrected)
Multiple Integrals
Week of February 10
16.1 Exercise 15 (to illustrate how to evaluate double sums in a do loop)
1 1
Estimate
1 C x eKy dx dy using the midpoint rule with the following number of squares of equal
0 0
size: 1, 4, 16, 64, 265, 1024.
First check the formula for the number of squares.
22 p $ p = 0 ..5
1, 4, 16, 64, 256, 1024
(1)
This means that there should be 2p subintervals in each direction.
Define the integrand function.
f x, y d
1 C x eKy
x, y / 1 C x eKy
This is what a for...do...end do loop looks like in Maple.
(2)
for p from 0 to 5
do
k d 2p :
1
k
k
> > evalf
f
i
1
j
1
K
, K
k 2k k 2k
:
k j = 1i = 1
print rectangles = 22 p , approximation = evalf %
end do:
2
rectangles = 1, approximation = 1.141606469
rectangles = 4, approximation = 1.143191426
rectangles = 16, approximation = 1.143534888
rectangles = 64, approximation = 1.143616875
rectangles = 256, approximation = 1.143637115
rectangles = 1024, approximation = 1.143642165
(3)
It turns out that Maple can evaluate this integral. The exact value is complicated so output is suppressed,
then displayed as a 10-digit approximation.
1 1
f x, y dx dy : evalf %
0 0
1.143643836
(4)
As an alternative, Maple will approximate the integral directly if decimals are used for the lower limits.
1
1
MapleApproximation =
f x, y dx dy
0.0 0.0
MapleApproximation = 1.143643841
(5)
Math 241, Lab 4 (corrected)
Multiple Integrals
Week of February 10
Your tasks for this lab are listed below.
Begin with the usual header:
Math 241
Lab 4
Name
Date
Save your file to your Documents Folder with the name JonesLab4Day1 (if your name is Jones).
1. 16.2 # 34.
2
Graph the solid that lies between the surfaces z = eKx cos x2 C y2 and z = 2 K x2 K y2 for
x % 1 and y % 1. Approximate the volume of this solid correct to 4 decimals.
2. 16.3 # 35.
Find the exact volume under the surface z = x3 y4 C x y2 and above the region bounded by the
curves y = x3 K x and y = x2 C x for x R 0.
Hint. First plot the two boundary curves over K2 % x % 2.2 using
plot x3 K x, x2 C x , x =K2 ..2.2 .
3. 16.5 # 22.
Find the mass, center of mass, and moments of inertia of the planar lamina that occupies the
region enclosed by the cardioid r = 1 C cos q if its planar density at x, y is
d x, y =
x2 C y2 .
4. 16.6 # 47.
Let E be the solid in the first octant bounded by the cylinder x2 C y2 = 1 and the planes
y = z, x = 0, and z = 0 with the mass density function d x, y, z = 1 C x C y C z. See the
picture. Find the exact values of the mass, the center of mass, and the moment of inertia about
the z-axis.
plots implicitplot3d x2 C y2 = 1, y = z, x = 0, z = 0 , x = 0 ..1, y = 0 ..1, z = 0 ..1, style = patchcontour,
color = red, blue, green, grey , orientation = K15, 60 , font = times, roman, 8 ,
tickmarks = 3, 3, 3
2