6. (Finiteness of integrals) (a) Suppose f is a function which is continuous on [a,b] (so a < b). By Theorem 3 in §5.2 in the book, the function f is integrable on [a,b]. That is to say, the definite integralJa (*)dæ exists (and is finite). Intuitively, this makes sense, since we're basically calculating the area of a bounded region. By the extreme value theorem (Theorem 3 in §4.1), f attains an absolute maximum and an absolute minimum somewhere on [a,b]. Suppose the absolute maximum is M and absolute minimum is m. Without knowing anything else about f, in terms of M,m,a, and b, what is the maximum that Ja J()da could be? What is the minimum that it could be? (b) Now we will consider what happens if the interval of integration is no longer finite. But before we get there, to begin, for practice, evaluate the integral 1000 1 (c) Now let's take it to the limit (literally). Let's consider the integral -dr (1) for some number r (you can just imagine r = 2). Intuitively, this integral represents the area under the graph ofy = F from x = 1 to o. However, as in homework 10, note that we have not yet defined what this expression (1) means, so we will define it here. In this case, this expression means: - dx := lim b00 1 -dx x" (2) Using this definition with r = 2, evaluate the integral dr (The answer should be a finite number)
6. (Finiteness of integrals) (a) Suppose f is a function which is continuous on [a,b] (so a < b). By Theorem 3 in §5.2 in the book, the function f is integrable on [a,b]. That is to say, the definite integralJa (*)dæ exists (and is finite). Intuitively, this makes sense, since we're basically calculating the area of a bounded region. By the extreme value theorem (Theorem 3 in §4.1), f attains an absolute maximum and an absolute minimum somewhere on [a,b]. Suppose the absolute maximum is M and absolute minimum is m. Without knowing anything else about f, in terms of M,m,a, and b, what is the maximum that Ja J()da could be? What is the minimum that it could be? (b) Now we will consider what happens if the interval of integration is no longer finite. But before we get there, to begin, for practice, evaluate the integral 1000 1 (c) Now let's take it to the limit (literally). Let's consider the integral -dr (1) for some number r (you can just imagine r = 2). Intuitively, this integral represents the area under the graph ofy = F from x = 1 to o. However, as in homework 10, note that we have not yet defined what this expression (1) means, so we will define it here. In this case, this expression means: - dx := lim b00 1 -dx x" (2) Using this definition with r = 2, evaluate the integral dr (The answer should be a finite number)
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Question
![6. (Finiteness of integrals)
(a) Suppose f is a function which is continuous on [a,b] (so a < b). By Theorem 3 in §5.2 in the book, the function
f is integrable on [a,b]. That is to say, the definite integralJa (*)dæ exists (and is finite). Intuitively, this
makes sense, since we're basically calculating the area of a bounded region. By the extreme value theorem
(Theorem 3 in §4.1), f attains an absolute maximum and an absolute minimum somewhere on [a,b]. Suppose
the absolute maximum is M and absolute minimum is m. Without knowing anything else about f, in terms of
M,m,a, and b, what is the maximum that Ja J()da could be? What is the minimum that it could be?
(b) Now we will consider what happens if the interval of integration is no longer finite. But before we get there,
to begin, for practice, evaluate the integral
1000
1
(c) Now let's take it to the limit (literally). Let's consider the integral
-dr
(1)
for some number r (you can just imagine r = 2). Intuitively, this integral represents the area under the graph
ofy = F from x = 1 to o. However, as in homework 10, note that we have not yet defined what this
expression (1) means, so we will define it here. In this case, this expression means:
- dx := lim
b00
1
-dx
x"
(2)
Using this definition with r = 2, evaluate the integral
dr
(The answer should be a finite number)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdb89fbe0-0a0e-4ccd-b147-bef9bc76b738%2Fc684e2a3-9a98-4766-95b3-6d47376abd1d%2Fwod3z8u.jpeg&w=3840&q=75)
Transcribed Image Text:6. (Finiteness of integrals)
(a) Suppose f is a function which is continuous on [a,b] (so a < b). By Theorem 3 in §5.2 in the book, the function
f is integrable on [a,b]. That is to say, the definite integralJa (*)dæ exists (and is finite). Intuitively, this
makes sense, since we're basically calculating the area of a bounded region. By the extreme value theorem
(Theorem 3 in §4.1), f attains an absolute maximum and an absolute minimum somewhere on [a,b]. Suppose
the absolute maximum is M and absolute minimum is m. Without knowing anything else about f, in terms of
M,m,a, and b, what is the maximum that Ja J()da could be? What is the minimum that it could be?
(b) Now we will consider what happens if the interval of integration is no longer finite. But before we get there,
to begin, for practice, evaluate the integral
1000
1
(c) Now let's take it to the limit (literally). Let's consider the integral
-dr
(1)
for some number r (you can just imagine r = 2). Intuitively, this integral represents the area under the graph
ofy = F from x = 1 to o. However, as in homework 10, note that we have not yet defined what this
expression (1) means, so we will define it here. In this case, this expression means:
- dx := lim
b00
1
-dx
x"
(2)
Using this definition with r = 2, evaluate the integral
dr
(The answer should be a finite number)
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