1. List the chambers and valves of the heart that blood must pass through whentraveling from the
superior/inferior vena cava to the aorta. Which valves are fitted with chordae tendineae and why? What effect
would damage to the papillary muscle in the left
366 El CHAPTER5 iNTEGRALS
n. CONCEPT CHECK
1. (a) 21:1 f (x?) Ax is an expression for a Riemann sum of a function f. x," is a point in the ith subinterval
[Jo-1, xi] and Ax is the length of the subintervals.
(b) See Figure 1 in Section 5.
378 PROBLEMS PLUS
14. Let x be the distance between the center of the disk and the surface of the liquid.
The wetted circular region has area 7rr2 me2 while the unexposed wetted region
(shaded in the diagram) has area 2 f; m dt, so the exposed wetted re
370 D CHAPTERS INTEGRALS
x dx) . . . . . . . .
18. [02? -does not ex15t smce the integrand has an innlte d1scont1nu1ty at x = 1.
19. /1 ~- does not exist since the integrand has an innite discontinuity at x = 1.
2 (2x + 3)4
20' f1 x + x3 + x5 dx =
CHAPTER 5 REVIEW 373
50. A1='h = % (2) (2) = 2, A2 = 'h = %(1) (1) = %, and since
y 2 v1 - x2 for 0 5 x S 1 represents a quarter-circle with radius 1,
A3 = nrz = %7r (1)2 = %. So
y =. \l 1 x2
51. By the Fundamental Theore
SECTIONS!) THE SUBSTITUTION RULE D 365
74.Letu=ex+1.Thendu=exdx,so/ 2 dx: d_u =ln|u|+C ln(x+1)+C.
75. Letu=x2+2x.Thendu=2(x+])dx,so/ x dx: li=1n1u|+C=11n|x2+2x|+C.
x2+2x u 2 2
76. Letu=cosx.Thendu=sinxdx,so/~dx = u =tan1u+C=tan'l(c
348 El CHAPTERS INTEGRALS
3x 2 0 2 3X 2 2x 2 3x 2
u 1 u 1 ul u 1 ul
41. = d : d d :- d d =>
gm /2 u2+1 [MIMI /0 u2+1 A u2+1 Jr/o 2+1
(2)01 1 d (3)02 1 d 4x2 1 9x2 1
1 :_ _._ h . 3 _ 2- 3-
gm (2x)2+l dx x)+(3x)2+1 dxm) 42+1+
SECTION 6.1 AREAS BETWEEN CURVES L7 391
27. An equation of the line through (0, O) and (2, l) is y = %x; through (0, O)
and (1, 6) isy = 6x; through (2, l) and (1, 6) isy = %x + 133.
A = fOl [(%x + 133) (6x)] dx + foz [(%x + 1%) ~ %x] dx
:f31(%x + 133) dx
SECTION 5.3 THE FUNDAMENTAL THEOREM OF CALCULUS E 351
d h(x) d a mo
54. / f (t) dt = f(t) dt + f(t) dt (where a is in the domain of f)
dx g(x) dx 160) a
d we . d W) , ,
LE ~ g fmdt + g; a ow = f(g(x)g (X)+f (1100)h (x)
= f(h (X)/1(X) f(g (X)g (x)
SECTION 5.2 THE DEFINITE INTEGRAL U 335
8. (21) Using the right endpoints to approximate If f (x) dx, we have
Z f(x,-)Ax :2[f(2) +f(4)+ f(6)] =2(8.3+2.3 10.5)= 0.2
(b) Using the left endpoints to approximate f: f (x) dx, we have
Z f(x,_|) Ax = 2[f
1. Dierentiating both sides of the equation x sin 7rx = fgz f (t) dt (using FTC] and the Chain Rule for the right
side) gives sin xx + ax cos 7n: = 2x f (x2). Letting x = 2 so that f (x2) = f '(4), we obtain
sin27z +27: c0s27r = 4f (4), so f
CHAPTERS REVIEW E] 371
3Zu=x2+l => x2=u1andxdx=0u,so
/ x:3+1dx=_/(ujgl) (0u)=%/(u1/2u'1/2)du
%(%u3/2 2u1/2) +C
%(x2 +1)3/2 (x2 + l)1/2 +C
33. From the graph, it appears that the area under the curve y = x J
between x = O and x = 4 is somewhat le
-we operate at the most basic level
-humans are better at categorizing than computers
3 levels of categorizing:
-Typical and Atypical affects the time it takes for u to categorize something
-its impossible to have vigilance (pay attention) for a long time
-change blindness: will lead to missed information
Two processes work together to make attention.
Bottom up processing: feature detection level. Looking at pics to see what differen
-With self regulation and instrumental conditioning we can have highly productive behavior
-Functional fixedness: you cannot come up with alternatives for everyday objects like a credit
card is used for money for example.
-Reliability: if y
Tutorial Problem #4
Clinical Problem: Diarrhea, Constipation and more
1. To understand the normal process of defecation.
2. Discuss the causes of constipation; how drugs (and other agents) overcome
1. Health Professionals Working With First Nations, Inuit, and Mtis
2. Six nations health care center website
3. Prenatal care.
University of Michigan Hea
336 D CHAPTERS INTEGRALS
13. In Maple, we use the command with (student) ; to load the sum and box commands, then
m: =middlesum ( sqrt (1+xA2) , x=1 . .2 , 10) ; which gives us the sum in summation notation, then
M:=evalf (m) ; which gives M10 8 1.8100141
SECTION 5.1 AREAS AND DISTANCES D 329
8. We can use the algorithm from Exercise 7 with X _MFN = 1, X _MAX = 2, and 1 / (RIGHT _ENDPOINT)2
instead of sin (RIGHT _ENDPOINT) in step 221. We nd that R10 _ 012 - 'V 0.4640,
1 3O 1 1 50 1