Question

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample. (a) True or False: $$ \sum_{j=1}^{10} \sum_{k=1}^{15} j k^{2}=\sum_{k=1}^{10} \sum_{j=1}^{15} j^{2} k $$ (b) True or False: $$ \sum_{j=1}^{m} \sum_{k=1}^{n} j^{2} k^{3}=\left(\sum_{j=1}^{m} j^{2}\right)\left(\sum_{k=1}^{n} k^{3}\right) $$ (c) True or False: $$ \sum_{j=1}^{m} \sum_{k=1}^{n} e^{j k^{2}}=\left(\sum_{j=1}^{m} e^{j}\right)\left(\sum_{k=1}^{n} e^{k^{2}}\right) $$ (d) True or False: If $f(x, y)$ is continuous on the rectangle $\mathcal{R}=\{(x, y) \mid a \leq x \leq b$ and $c \leq y \leq d\}$, then $\iint_{\mathcal{R}} f(x, y) d A$ exists. (e) True or False: If $f(x, y)$ is continuous on the rectangle $\mathcal{R}=\{(x, y) \mid a \leq x \leq b$ and $c \leq y \leq d\}$, then $\iint_{\mathcal{P}} f(x, y) d A=\int_{a}^{b} \int_{c}^{d} f(x, y) d x d y$ (f) True or False: If $f(x, y)$ is continuous on the rectangle $\mathcal{R}=\{(x, y) \mid a \leq x \leq b$ and $c \leq y \leq d\}$, then $\iint_{\mathbb{R}} f(x, y) d A=\int_{a}^{b} \int_{c}^{d} f(x, y) d y d x .$ (g) True or False: If $f(x, y)$ is continuous on the rectangle $\mathcal{R}=\{(x, y) \mid a \leq x \leq b$ and $c \leq y \leq d\}$, then $\int_{b}^{a} \int_{d}^{c} f(x, y) d y d x=\int_{a}^{b} \int_{c}^{d} f(x, y) d y d x$. (h) True or False: If $f(x, y)$ is continuous on the rectangle $\mathcal{R}=\{(x, y) \mid a \leq x \leq b$ and $c \leq y \leq d\}$, then $\int_{c}^{d} \int_{a}^{b} f(x, y) d x d y=\int_{a}^{b} \int_{c}^{d} f(x, y) d y d x$.

Name: Problem 1, Chapter 13: Calculus: Early Transcendentals
Uploaded: 2022-04-10T19:25:32-08:00
Duration: 5 min 1 s
Channel: Lucas Finney

   True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.
(a) True or False:
$$
\sum_{j=1}^{10} \sum_{k=1}^{15} j k^{2}=\sum_{k=1}^{10} \sum_{j=1}^{15} j^{2} k
$$
(b) True or False:
$$
\sum_{j=1}^{m} \sum_{k=1}^{n} j^{2} k^{3}=\left(\sum_{j=1}^{m} j^{2}\right)\left(\sum_{k=1}^{n} k^{3}\right)
$$
(c) True or False:
$$
\sum_{j=1}^{m} \sum_{k=1}^{n} e^{j k^{2}}=\left(\sum_{j=1}^{m} e^{j}\right)\left(\sum_{k=1}^{n} e^{k^{2}}\right)
$$
(d) True or False: If $f(x, y)$ is continuous on the rectangle $\mathcal{R}=\{(x, y) \mid a \leq x \leq b$ and $c \leq y \leq d\}$, then $\iint_{\mathcal{R}} f(x, y) d A$ exists.
(e) True or False: If $f(x, y)$ is continuous on the rectangle $\mathcal{R}=\{(x, y) \mid a \leq x \leq b$ and $c \leq y \leq d\}$, then $\iint_{\mathcal{P}} f(x, y) d A=\int_{a}^{b} \int_{c}^{d} f(x, y) d x d y$
(f) True or False: If $f(x, y)$ is continuous on the rectangle $\mathcal{R}=\{(x, y) \mid a \leq x \leq b$ and $c \leq y \leq d\}$, then $\iint_{\mathbb{R}} f(x, y) d A=\int_{a}^{b} \int_{c}^{d} f(x, y) d y d x .$
(g) True or False: If $f(x, y)$ is continuous on the rectangle $\mathcal{R}=\{(x, y) \mid a \leq x \leq b$ and $c \leq y \leq d\}$, then $\int_{b}^{a} \int_{d}^{c} f(x, y) d y d x=\int_{a}^{b} \int_{c}^{d} f(x, y) d y d x$.
(h) True or False: If $f(x, y)$ is continuous on the rectangle $\mathcal{R}=\{(x, y) \mid a \leq x \leq b$ and $c \leq y \leq d\}$, then $\int_{c}^{d} \int_{a}^{b} f(x, y) d x d y=\int_{a}^{b} \int_{c}^{d} f(x, y) d y d x$.

Calculus: Early Transcendentals

Michael Sullivan,… 1st Edition

Chapter 13, Problem 1

Instant Answer

Solved by Expert Lucas Finney

04/10/2022

Step 1

The first sum is $\sum_{j=1}^{10} \sum_{k=1}^{15} j k^{2}$ and the second sum is $\sum_{k=1}^{10} \sum_{j=1}^{15} j^{2} k$. We can see that in both sums, we are summing over the same range of values for $j$ and $k$, and the terms in the sum are the same, just Show more…

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True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample. (a) True or False: $$ \sum_{j=1}^{10} \sum_{k=1}^{15} j k^{2}=\sum_{k=1}^{10} \sum_{j=1}^{15} j^{2} k $$ (b) True or False: $$ \sum_{j=1}^{m} \sum_{k=1}^{n} j^{2} k^{3}=\left(\sum_{j=1}^{m} j^{2}\right)\left(\sum_{k=1}^{n} k^{3}\right) $$ (c) True or False: $$ \sum_{j=1}^{m} \sum_{k=1}^{n} e^{j k^{2}}=\left(\sum_{j=1}^{m} e^{j}\right)\left(\sum_{k=1}^{n} e^{k^{2}}\right) $$ (d) True or False: If $f(x, y)$ is continuous on the rectangle $\mathcal{R}=\{(x, y) \mid a \leq x \leq b$ and $c \leq y \leq d\}$, then $\iint_{\mathcal{R}} f(x, y) d A$ exists. (e) True or False: If $f(x, y)$ is continuous on the rectangle $\mathcal{R}=\{(x, y) \mid a \leq x \leq b$ and $c \leq y \leq d\}$, then $\iint_{\mathcal{P}} f(x, y) d A=\int_{a}^{b} \int_{c}^{d} f(x, y) d x d y$ (f) True or False: If $f(x, y)$ is continuous on the rectangle $\mathcal{R}=\{(x, y) \mid a \leq x \leq b$ and $c \leq y \leq d\}$, then $\iint_{\mathbb{R}} f(x, y) d A=\int_{a}^{b} \int_{c}^{d} f(x, y) d y d x .$ (g) True or False: If $f(x, y)$ is continuous on the rectangle $\mathcal{R}=\{(x, y) \mid a \leq x \leq b$ and $c \leq y \leq d\}$, then $\int_{b}^{a} \int_{d}^{c} f(x, y) d y d x=\int_{a}^{b} \int_{c}^{d} f(x, y) d y d x$. (h) True or False: If $f(x, y)$ is continuous on the rectangle $\mathcal{R}=\{(x, y) \mid a \leq x \leq b$ and $c \leq y \leq d\}$, then $\int_{c}^{d} \int_{a}^{b} f(x, y) d x d y=\int_{a}^{b} \int_{c}^{d} f(x, y) d y d x$.

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Key Concepts

Iterated Summation and Commutativity

This concept focuses on the property that when summing over two indices, the summation order can typically be interchanged. However, one must be cautious because the individual summands themselves might not be symmetric and the limits of summation might be different. In problems like these, interchanging the summation order does not necessarily guarantee that two differently ordered sums will yield the same result if the expressions being summed have different forms.

Separable Sums and Products

This key idea pertains to when a double summation of a product of functions can be factored into the product of two single summations. If the summand is the product of a function solely of one index and another function solely of the other index, then the iterated sum can be written as a product of separate sums. However, when the exponents or functions depend in a non?separable way on the indices, the equality between the double sum and the product of sums may fail.

Continuous Functions and Riemann Integrability

The notion that every continuous function on a closed and bounded rectangle is Riemann integrable is fundamental in analysis. This idea ensures that the double integral of such functions exists because continuous functions on compact sets are integrable. This concept underlies the justification that when a function is continuous on a rectangle, its double integral over that region is well defined.

Fubini's Theorem

Fubini's Theorem is a central result in multivariable calculus that allows one to compute a double integral as an iterated integral. It guarantees that if a function is continuous (or more generally, integrable under certain conditions) on a product set, then the order of integration can be interchanged. This theorem is crucial for evaluating double integrals and justifies the equality of different iterated integrals over rectangular regions.

Order of Integration and Reversal of Limits

This concept addresses the impact of reversing the order or limits of integration in iterated integrals. While Fubini's Theorem allows for the reordering of integration, reversing the integration limits introduces a sign change, which must be accounted for. Integrals taken from the upper limit to the lower limit are typically defined as the negative of the integral from the lower limit to the upper limit, emphasizing the importance of correctly handling the direction of integration.

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Transcript

00:01 For this problem, we are first asked to determine if the following statement is true or false, that the sum from j equals 1 up to 10 of the sum from k equals 1 up to 15 of j times k squared equals the sum from k equals 1 up to 10 times the or of the sum from j equals 1 up to 15 of k.

00:24 Or pardon me, sum from j equals 1 to 15 of j squared k.

00:29 So if we think about that first series, we would have 1 times 1 squared plus 1 times 2 squared, so 1 times 4, plus 1 times 3 squared, so plus 1 times 9 plus dot, dot, dot, dot.

00:48 And then we would expect to have plus 2 times 1 plus 2 times 4 plus 2 times 9 plus dot, dot, dot, dot, and we'd go from 1 to 15.

00:59 And with the second, we'd have one square times one, so one, plus two square times one, so plus four, plus nine, plus dot, dot, dot.

01:10 And we can see that the thing that's getting squared, we are summing over first, and then we're summing over the constant coefficient.

01:23 So effectively, we can see that, yes, it's true that these will be equal.

01:27 Then we have true or false the sum from j equals 1 to m of the sum from k equals 1 to n of j squared times k cubed equals the product of those sums that is quite trivially not going to be true that's false then for we have true or false the sum from j equals 1 up to n the sum from k equals 1 up to n of e times j k squared equals the sum from j equals one to m of e to the power of j times the sum from k equals one up to n of e to the power of k squared so this is also going to be false because we would effectively get just from expanding out that first sum we would have e times the sum from k equals one up to n of e to the power of k squared, plus e to the power of 1 times the sum from k equals 1, or excuse me, plus e to power of 2 times the sum from k equals 1 up to n of e to power of k squared plus dot dot dot dot.

02:35 That is quite clearly not going to lead, oh, actually, let me step back for a second here, because i just realized that actually that is going to be equivalent.

02:50 We are iterating over j and wait.

02:57 No, it's definitely not going to be equivalent.

02:59 I take that back.

03:00 It's false.

03:02 For d, we have true or false...

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