As a hardworking student, plagued by too much homework, you spend all night doing math homework. By 6am, you imagine yourself to be a region bounded by Y 7x² - X = 0 x = 1 y = 0 As you grow more and more tired, the world begins to spin around you. However, according to Newton, there is no difference between the world spinning around you, and you spinning around the world. Unfortunately, you are so tired that you think the world is the x-axis. What is the volume of the solid you (the region) create by spinning about the x-axis?
As a hardworking student, plagued by too much homework, you spend all night doing math homework. By 6am, you imagine yourself to be a region bounded by Y 7x² - X = 0 x = 1 y = 0 As you grow more and more tired, the world begins to spin around you. However, according to Newton, there is no difference between the world spinning around you, and you spinning around the world. Unfortunately, you are so tired that you think the world is the x-axis. What is the volume of the solid you (the region) create by spinning about the x-axis?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![As a hardworking student, overwhelmed by too much homework, you spend all night doing math assignments. By 6am, you imagine yourself to be a region bounded by the following equations:
\[ y = 7x^2 \]
\[ x = 0 \]
\[ x = 1 \]
\[ y = 0 \]
As you grow more and more tired, the world begins to spin around you. However, according to Newton, there is no difference between the world spinning around you and you spinning around the world. Unfortunately, you are so tired that you think the world is the x-axis. What is the volume of the solid you (the region) create by spinning about the x-axis?
**Explanation for Graph or Diagram:**
The task involves finding the volume of a solid of revolution. The region bounded by the curve \( y = 7x^2 \), the lines \( x = 0 \), \( x = 1 \), and \( y = 0 \) is rotated around the x-axis to create the solid.
To find the volume, use the disk method formula:
\[ V = \pi \int_a^b [f(x)]^2 \, dx \]
For this scenario:
- \( f(x) = 7x^2 \)
- The limits of integration are from \( x = 0 \) to \( x = 1 \).
Thus, the volume \( V \) will be:
\[ V = \pi \int_0^1 (7x^2)^2 \, dx \]
\[ V = \pi \int_0^1 49x^4 \, dx \]
Evaluating this integral will yield the volume of the solid.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd76112a2-3370-45d7-a044-1f0297120ded%2F32e1160a-d41e-4c05-886a-580cc1f4e1d2%2Farc3va_processed.jpeg&w=3840&q=75)
Transcribed Image Text:As a hardworking student, overwhelmed by too much homework, you spend all night doing math assignments. By 6am, you imagine yourself to be a region bounded by the following equations:
\[ y = 7x^2 \]
\[ x = 0 \]
\[ x = 1 \]
\[ y = 0 \]
As you grow more and more tired, the world begins to spin around you. However, according to Newton, there is no difference between the world spinning around you and you spinning around the world. Unfortunately, you are so tired that you think the world is the x-axis. What is the volume of the solid you (the region) create by spinning about the x-axis?
**Explanation for Graph or Diagram:**
The task involves finding the volume of a solid of revolution. The region bounded by the curve \( y = 7x^2 \), the lines \( x = 0 \), \( x = 1 \), and \( y = 0 \) is rotated around the x-axis to create the solid.
To find the volume, use the disk method formula:
\[ V = \pi \int_a^b [f(x)]^2 \, dx \]
For this scenario:
- \( f(x) = 7x^2 \)
- The limits of integration are from \( x = 0 \) to \( x = 1 \).
Thus, the volume \( V \) will be:
\[ V = \pi \int_0^1 (7x^2)^2 \, dx \]
\[ V = \pi \int_0^1 49x^4 \, dx \]
Evaluating this integral will yield the volume of the solid.
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