CALCULUS REVISITED
PART 2
A S e l f -Study .Course
STUDY G U I D E
Block 5
Multiple Integration
H e r b e r t I. Gross Senior Lecturer C e n t e r f o r Advanced E n g i n e e r i n g Study
Massachusetts I n s t i t u t e of
Technology
Copyright @ 1972 by
Massachusetts Institute of Technology Cambridge, Massachusetts All rights reserved. No part of this book may be reproduced in any form or by any means without permission in writing from the Center for Advanced Engineering Study, M.I.T. CONTENTS
Study Guide
Block 5: M u l t i p l e I n t e g r a t i o n
Pretest
U n i t 1: Double Sums
U n i t 2: The Double Sum a s an I t e r a t e d I n t e g r a l
U n i t 3: Change of V a r i a b l e s i n M u l t i p l e I n t e g r a l s
U n i t 4 : Volumes and Masses of More General S o l i d s
U n i t 5: More on P o l a r C o o r d i n a t e s
Unit 6 : S u r f a c e Area
U n i t 7: L i n e I n t e g r a l s
U n i t 8 : Green's Theorem
Quiz
Solutions
Block 5: M u l t i p l e I n t e g r a t i o n
Pretest
U n i t 1: Double Sums
U n i t 2: The Double Sum a s an I t e r a t e d I n t e g r a l
U n i t 3 : Change of V a r i a b l e s i n M u l t i p l e I n t e g r a l s
U n i t 4 : Volumes and Masses of More General S o l i d s
U n i t 5: More on P o l a r C o o r d i n a t e s
U n i t 6 : S u r f a c e Area
U n i t 7: L i n e I n t e g r a l s
U n i t 8 : G r e e n ' s Theorem
Quiz
Study Guide BLOCK 5: MULTIPLE INTEGRATION Study Guide
Block 5: M u l t i p l e I n t e g r a t i o n
Pretest
1. Compute t h e sum
2. Evaluate
3 . By changing t h e o r d e r of i n t e g r a t i o n , e v a l u a t e
[
Y
dy dx.
Use p o l a r c o o r d i n a t e s t o compute
4.
2
2-(x
+ Y2)
dy dx.
5. Find t h e mass of t h e s o l i d S i f S i s given by t h e equation
x 2 + y 2 + z2 < 1 and t h e d e n s i t y of S a t each p o i n t is
numerically e q u a l t o t h e d i s t a n c e of t h a t p o i n t from t h e o r i g i n .
6. Find t h e s u r f a c e a r e a of t h e region S where S i s t h a t p o r t i o n
'
of z = 1 - x2
-
y2 f o r which z > 0.
7. Find t h e volume of t h e region which i s bounded between t h e two
2
2
x
e l l i p t i c p a r a b o l o i d s z = x2 + gy2 and z = 18
y
8. L e t c be t h e s q u a r e d e f i n e d by t h e e q u a t i o n
Compute
56
ydx
-
xdy.
-
-
I xl
+ 1 y1
.
= 1.
Study Guide Block 5: Multiple Integration Unit 1 : Double Sums
1.
Overview In the same way that certain types of infinite sums (i.e., definite integrals) can be identified with inverse differentiation (The First Fundamental Theorem of Integral Calculus), certain types of "multiple infinite sums" can be identified with inverse partial differentiation. In this unit, the lecture is primarily concerned with introducing the notion of a double infinite sum (the generali- zation to multiple infinite sums being rather straight-forward). The lecture introduces the double sum in terms of finding the mass of a "thin plate" which has a density that varies from point to point on the plate. The exercises serve two purposes. On the one hand, the first few exercises are used to help you become more familiar with the notion of a double sum, and to help you get an idea of the computational properties bf such sums. Exercise 5.1.4(L) is used to show the subleties that occur when we make the transition from double finite sums to double infinite sums. The problems are similar (but in some ways a bit more sophisticated) to what happened in the study of calculus of a single real variable when we went from adding arbitrarily large (but finite) numbers of terms to adding infinitely many terms; and we investigated how certain "trivial" properties for finite sums became "luxuries" for infinite sums. The remainder of the exercises "refine" and amplify the computational techniques which are introduced in the lecture. Study Guide Block 5: Multiple Integration Unit 1: Double Sums 2.
Lecture 5.010 Study Guide Block 5: Multiple Integration Unit 1: Double Sums 3.
Exercises :
5.1.1(L) The expression is an abbreviation for Similarly, is an abbreviation for Show that and 5
are equal, and discuss the nature of the terms that make up both sums. Apply the results of Exercise 5.1.1(L) to compute each of the following double sums. (continued on next page) 5.1.3
Study Guide Block 5: Multiple Integration Unit 1: Double Sums 5.1.2 continued x
4
b. i=1
3 z i j
j=1
and j=1
i=1 5.1.3 (L) Verify each of the following. where c is a constant, independent of i and j. .
Use part (b) of this exercise to check the result stated in part
(b) of the previous exercise. e. Use the result of part (d) to check part (c) of the previous exercise. m
The symbol
[
n+m i=l
i=1
a
j=1
z
m
aij, by definition, denotes g l l i m t a i j l =
j=1
i=l m+m j=1
.
~imilarly,
j=1
(continued on next page) 5.1.4 e a i j denotes :=I
Study Guide Block 5: Multiple Integration Unit 1: Double Sums 5.1.4 (L) continued [qlim
5
aij
]
.
We now define ai, , for all positive inten-tw
j=1
i=l
gral values of i and j, by lim
rn-
1, if i = j -1, if i = j - 1
0, otherwise. Compute
C
in this example
(j = 2,3,...) aij and
C
Caij
aij and show. in particular, that 'C
Caij'
Consider the thin plate in the shape of a unit square with vertices at (0,0), (0,1), (1,0), and (1,l). Suppose that at each point (x,y) in the plate the density of the plate is given by 2
2
p(x,y) = x + y
.
a. Mimic our procedure
for the mass of the
the number of equal
to (0,l) and n, the
partitioned. in the lecture to find upper and lower bounds plate as functions of m and n where m denotes parts into which we partition the segment (0,O) number of parts into which (0,O) to (1,O) is b. Use the result of (a) with n = m = lo6 to show that to five significant figures the mass of the plate is 0.66667.
c. Let m and n each approach infinity in part (a) and thus deduce the exact mass of the plate. d. The solid S is the parallelepiped whose base is the unit square
with vertices (0,O), (0,l), (1,O), and (Ill) and whose top is the
surface z = x2 + y2
Find the volume of S.
.
Study Guide Block 5: Multiple Integration Unit 1: Double Sums 5.1.6 Suppose R is the same thin-plate as in the previous exercise, but
the density of the plate at the point (x,y) is now given by
p(x,y) = xy. Find the mass of R.
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