12.2 – Iterated Integrals 1. The following integrals are both nonsense. Explain what is wrong with each of them. 5 5x !! f(x,y) dx dy rsin (x) f(x,y) dy dx -x Jx
12.2 – Iterated Integrals 1. The following integrals are both nonsense. Explain what is wrong with each of them. 5 5x !! f(x,y) dx dy rsin (x) f(x,y) dy dx -x Jx
Calculus: Early Transcendentals
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Chapter1: Functions And Models
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Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![# 12.2 – Iterated Integrals
## 1. The following integrals are both nonsense. Explain what is wrong with each of them.
### Integral 1:
\[
\int_{0}^{1} \int_{0}^{5x} f(x, y) \, dx \, dy
\]
**Explanation:**
In this iterated integral, the limits of integration for \(x\) are from \(0\) to \(5x\). This doesn't make sense because the upper limit for \(x\) is expressed in terms of \(x\) itself, which is not valid for definite integrals. The limits must be constants or functions of the outer variable \(y\).
### Integral 2:
\[
\int_{-x}^{x} \int_{0}^{\sin(x)} f(x, y) \, dy \, dx
\]
**Explanation:**
Here, the integration with respect to \(y\) has limits from \(0\) to \(\sin(x)\). However, the limits for the outer integral with respect to \(x\) are from \(-x\) to \(x\), which again involve the variable of integration itself. This is invalid because limits should be constants or functions of \(y\), the outer integral variable in this context. Additionally, having \(-x\) as a lower limit is problematic because \(x\) appears in its own limits of integration.
In both cases, the formulations of the integral limits are incorrect, leading to nonsensical iterated integrals.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F415836e0-9819-4f43-92ec-1c78267e69f7%2F9c1085e6-f21f-4a59-a619-3f04415ec32c%2Fzw8to08_processed.png&w=3840&q=75)
Transcribed Image Text:# 12.2 – Iterated Integrals
## 1. The following integrals are both nonsense. Explain what is wrong with each of them.
### Integral 1:
\[
\int_{0}^{1} \int_{0}^{5x} f(x, y) \, dx \, dy
\]
**Explanation:**
In this iterated integral, the limits of integration for \(x\) are from \(0\) to \(5x\). This doesn't make sense because the upper limit for \(x\) is expressed in terms of \(x\) itself, which is not valid for definite integrals. The limits must be constants or functions of the outer variable \(y\).
### Integral 2:
\[
\int_{-x}^{x} \int_{0}^{\sin(x)} f(x, y) \, dy \, dx
\]
**Explanation:**
Here, the integration with respect to \(y\) has limits from \(0\) to \(\sin(x)\). However, the limits for the outer integral with respect to \(x\) are from \(-x\) to \(x\), which again involve the variable of integration itself. This is invalid because limits should be constants or functions of \(y\), the outer integral variable in this context. Additionally, having \(-x\) as a lower limit is problematic because \(x\) appears in its own limits of integration.
In both cases, the formulations of the integral limits are incorrect, leading to nonsensical iterated integrals.
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