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Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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Calculus 2 HW
Please calculate the improper

Transcribed Image Text:Certainly! Here are the transcriptions of the integrals shown in the image:
9. \(\int_{4}^{\infty} e^{-y/2} \, dy\)
11. \(\int_{2\pi}^{\infty} \sin \theta \, d\theta\)
These integrals are both improper, extending to infinity as one of their limits. The first represents the integration of an exponential decay function, while the second involves the sine trigonometric function.
- **Integral 9** requires evaluating the behavior of the exponential function, which generally decreases towards zero as \(y\) increases.
- **Integral 11** involves the periodic sine function, which oscillates indefinitely, requiring consideration of its convergence or divergence over the specified interval.
![### Transcription
#### Integrals
5. Evaluate the integral:
\[
\int_{3}^{\infty} \frac{1}{(x - 2)^{3/2}} \, dx
\]
6. Evaluate the integral:
\[
\int_{0}^{\infty} \frac{1}{\sqrt[4]{1 + x}} \, dx
\]
### Description
The image shows two mathematical integrals, which are common in calculus, where the goal is to determine the area under specific curves. Each integral has its respective bounds and integral function provided.
- **Integral 5**:
- The function is \(\frac{1}{(x - 2)^{3/2}}\), which potentially involves an improper integral due to its bounds going from 3 to \(\infty\). The term \((x - 2)^{3/2}\) implies a power function where \(x\) is offset by 2.
- **Integral 6**:
- The function is \(\frac{1}{\sqrt[4]{1 + x}}\). The interval of integration starts from 0 and extends to \(\infty\). The expression \(\sqrt[4]{1 + x}\) refers to the fourth root of the term \(1 + x\), indicating a radical function that grows at a rate determined by that root.](https://content.bartleby.com/qna-images/question/0e691e7d-3aa6-4d87-8ff6-0afc8b27fff3/efd08bee-994e-4fbd-ad39-2279dfef2e0d/1eos70di_thumbnail.jpeg)
Transcribed Image Text:### Transcription
#### Integrals
5. Evaluate the integral:
\[
\int_{3}^{\infty} \frac{1}{(x - 2)^{3/2}} \, dx
\]
6. Evaluate the integral:
\[
\int_{0}^{\infty} \frac{1}{\sqrt[4]{1 + x}} \, dx
\]
### Description
The image shows two mathematical integrals, which are common in calculus, where the goal is to determine the area under specific curves. Each integral has its respective bounds and integral function provided.
- **Integral 5**:
- The function is \(\frac{1}{(x - 2)^{3/2}}\), which potentially involves an improper integral due to its bounds going from 3 to \(\infty\). The term \((x - 2)^{3/2}\) implies a power function where \(x\) is offset by 2.
- **Integral 6**:
- The function is \(\frac{1}{\sqrt[4]{1 + x}}\). The interval of integration starts from 0 and extends to \(\infty\). The expression \(\sqrt[4]{1 + x}\) refers to the fourth root of the term \(1 + x\), indicating a radical function that grows at a rate determined by that root.
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