A tank on the wing of a jet aircraft is formed by revolving the region bounded by the graph of the function shown below and the x-axis (0 ≤ x ≤ 4) about the x-axis, where x and y are measured in meters. Use a graphing utility to graph the function. Find the volume of the tank. y=1/64x2 square root of 4-x
A tank on the wing of a jet aircraft is formed by revolving the region bounded by the graph of the function shown below and the x-axis (0 ≤ x ≤ 4) about the x-axis, where x and y are measured in meters. Use a graphing utility to graph the function. Find the volume of the tank. y=1/64x2 square root of 4-x
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
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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A tank on the wing of a jet aircraft is formed by revolving the region bounded by the graph of the function shown below and the x-axis (0 ≤ x ≤ 4) about the x-axis, where x and y are measured in meters. Use a graphing utility to graph the function. Find the volume of the tank. y=1/64x2 square root of 4-x
![**Problem Statement:**
A tank on the wing of a jet aircraft is formed by revolving the region bounded by the graph of the function shown below and the x-axis \((0 \leq x \leq 4)\) about the x-axis, where \(x\) and \(y\) are measured in meters. Use a graphing utility to graph the function. Find the volume of the tank.
\( y = \frac{1}{64} x^2 (4 - x) \)
[ _Blank space for calculation or answer_ ] m\(^3\)
**Explanation of the Function:**
The function given, \( y = \frac{1}{64} x^2 (4 - x) \), represents the shape of the region to be rotated. This function is a quadratic expression dependent on \( x \), where \( x \) ranges from 0 to 4 meters.
**Instructions:**
1. Use a graphing tool to plot the curve. You'll observe a parabola-like shape, as the equation is quadratic.
2. The volume of the tank formed by revolving this region about the x-axis can be found using the disk method in calculus.
3. Calculate the definite integral of the function, considering the revolution, to find the exact volume.
**Visualization:**
- Graph the function to better understand the area being revolved.
- Visualize the region between the curve and the x-axis from \( x = 0 \) to \( x = 4 \).
**Additional Tips:**
- The formula for calculating the volume using the disk method is:
\[
V = \pi \int_{a}^{b} [f(x)]^2 \, dx
\]
Substitute \( f(x) \) with the given function and integrate from 0 to 4.
- Ensure the graphing tool accurately represents the function to avoid errors in visualization.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faf2a9245-87db-4e54-b147-768ccba693e7%2Fab9a3678-1ae8-49aa-89cb-1f682a8135be%2Fxqh89b4_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
A tank on the wing of a jet aircraft is formed by revolving the region bounded by the graph of the function shown below and the x-axis \((0 \leq x \leq 4)\) about the x-axis, where \(x\) and \(y\) are measured in meters. Use a graphing utility to graph the function. Find the volume of the tank.
\( y = \frac{1}{64} x^2 (4 - x) \)
[ _Blank space for calculation or answer_ ] m\(^3\)
**Explanation of the Function:**
The function given, \( y = \frac{1}{64} x^2 (4 - x) \), represents the shape of the region to be rotated. This function is a quadratic expression dependent on \( x \), where \( x \) ranges from 0 to 4 meters.
**Instructions:**
1. Use a graphing tool to plot the curve. You'll observe a parabola-like shape, as the equation is quadratic.
2. The volume of the tank formed by revolving this region about the x-axis can be found using the disk method in calculus.
3. Calculate the definite integral of the function, considering the revolution, to find the exact volume.
**Visualization:**
- Graph the function to better understand the area being revolved.
- Visualize the region between the curve and the x-axis from \( x = 0 \) to \( x = 4 \).
**Additional Tips:**
- The formula for calculating the volume using the disk method is:
\[
V = \pi \int_{a}^{b} [f(x)]^2 \, dx
\]
Substitute \( f(x) \) with the given function and integrate from 0 to 4.
- Ensure the graphing tool accurately represents the function to avoid errors in visualization.
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