Please explain steps in details: Density of water is approximately 1000 kg/m^3 and gravity is 9.81 m/s^2. The function y=x^2 from x = 0 to 2 is rotated about the y-axis. 1) Find the volume of the resulting solid 2) If the resulting solid is filled with water, find the work done in emptying it - assume all units of measurement are in meters. Do we subtract? 3) Find the surface area of the resulting shape (set up the integral, and find a numerical approximation)
Please explain steps in details: Density of water is approximately 1000 kg/m^3 and gravity is 9.81 m/s^2. The function y=x^2 from x = 0 to 2 is rotated about the y-axis. 1) Find the volume of the resulting solid 2) If the resulting solid is filled with water, find the work done in emptying it - assume all units of measurement are in meters. Do we subtract? 3) Find the surface area of the resulting shape (set up the integral, and find a numerical approximation)
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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![### Detailed Steps for Calculating Rotation of a Solid
**Given:**
- Density of water: \(1000 \, \text{kg/m}^3\)
- Gravity: \(9.81 \, \text{m/s}^2\)
**Function and Rotation:**
- The function \( y = x^2 \) from \( x = 0 \) to \( x = 2 \) is rotated about the y-axis.
#### Tasks:
1. **Find the Volume of the Resulting Solid:**
- Use the disk or washer method to set up the integral for volume. Integrate with respect to \( x \) from 0 to 2.
2. **Calculate the Work Done in Emptying the Solid Filled with Water:**
- Assume all units are in meters.
- Identify how to set up the integral representing work using the formula:
\[
\text{Work} = \int_{a}^{b} \rho \, g \, A(y) \, h(y) \, dy
\]
Where \(\rho\) is the density, \(g\) is gravity, \(A(y)\) is the area at height \(y\), and \(h(y)\) is the height the water needs to be lifted.
3. **Find the Surface Area of the Resulting Shape:**
- Set up the integral to find the surface area.
- Perform the integration to find a numerical approximation.
Include diagrams and calculations for clarity, if applicable, and ensure all steps show how the mathematical concepts are applied.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F272afd08-fa1b-4f9b-918c-9c781af320ae%2Fc85edd6e-8d94-4319-9a6c-995c7467a9ad%2Fwgdlx7_processed.png&w=3840&q=75)
Transcribed Image Text:### Detailed Steps for Calculating Rotation of a Solid
**Given:**
- Density of water: \(1000 \, \text{kg/m}^3\)
- Gravity: \(9.81 \, \text{m/s}^2\)
**Function and Rotation:**
- The function \( y = x^2 \) from \( x = 0 \) to \( x = 2 \) is rotated about the y-axis.
#### Tasks:
1. **Find the Volume of the Resulting Solid:**
- Use the disk or washer method to set up the integral for volume. Integrate with respect to \( x \) from 0 to 2.
2. **Calculate the Work Done in Emptying the Solid Filled with Water:**
- Assume all units are in meters.
- Identify how to set up the integral representing work using the formula:
\[
\text{Work} = \int_{a}^{b} \rho \, g \, A(y) \, h(y) \, dy
\]
Where \(\rho\) is the density, \(g\) is gravity, \(A(y)\) is the area at height \(y\), and \(h(y)\) is the height the water needs to be lifted.
3. **Find the Surface Area of the Resulting Shape:**
- Set up the integral to find the surface area.
- Perform the integration to find a numerical approximation.
Include diagrams and calculations for clarity, if applicable, and ensure all steps show how the mathematical concepts are applied.
Transcribed Image Text:The image represents a coordinate system with a parabola and a horizontal line:
1. **Graphs:**
- **Parabola**: The equation is \( y = x^2 \). This parabola opens upwards, and its vertex is at the origin (0,0).
- **Horizontal Line**: The equation is \( y = 4 \). This line is parallel to the x-axis and intersects the parabola.
2. **Points and Intersections:**
- Intersection points between the parabola and the line are marked at (2,4) and (-2,4).
- A generic point on the parabola is labeled as \( (x, y) \).
3. **Annotations:**
- The vertical distance from the x-axis up to the line \( y = 4 \) is marked as 4 units.
- The distance from any point on the parabola to the line \( y = 4 \) along the vertical direction is marked as \( 4 - y \).
- There is a horizontal arrow pointing to a disc, labeled “disc of radius x,” on the parabola's interior.
4. **Additional Details:**
- The points where the parabola intersects the line \( y = 4 \) are highlighted, and their coordinates are given.
- Vertical lines are drawn from the x-axis to illustrate the height 4, and dashed lines show the level of \( y = 4 \).
This diagram might be used to illustrate concepts related to the intersection of a parabola with a line, as well as discussing the properties of shapes and distances in a coordinate system.
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