Find the volume of the solid obtained by rotating the region bounded by the curves y = y 0 about the y-axis. Give an exact answer in terms of . = 3x T ²2 and
Find the volume of the solid obtained by rotating the region bounded by the curves y = y 0 about the y-axis. Give an exact answer in terms of . = 3x T ²2 and
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**Problem Statement:**
Find the volume of the solid obtained by rotating the region bounded by the curves \( y = 3x - x^2 \) and \( y = 0 \) about the \( y \)-axis. Give an exact answer in terms of \( \pi \).
**Explanation:**
To solve this problem, use the method of cylindrical shells. First, identify the region bounded by the equations in the xy-plane. The curve \( y = 3x - x^2 \) is a downward-facing parabola with roots at \( x = 0 \) and \( x = 3 \). The line \( y = 0 \) is the x-axis.
The volume of the solid is given by the integral:
\[
V = 2\pi \int_{a}^{b} x \cdot f(x) \, dx
\]
Here, \( f(x) = 3x - x^2 \), and the limits of integration \( a \) and \( b \) are the x-values where the parabola intersects the x-axis, \( 0 \) and \( 3 \).
Calculate the integral:
\[
V = 2\pi \int_{0}^{3} x(3x - x^2) \, dx
= 2\pi \int_{0}^{3} (3x^2 - x^3) \, dx
= 2\pi \left[ x^3 - \frac{x^4}{4} \right]_{0}^{3}
\]
Evaluate the integral:
\[
= 2\pi \left[ (27 - \frac{81}{4}) - (0 - 0) \right]
= 2\pi \left[ \frac{108}{4} - \frac{81}{4} \right]
= 2\pi \left[ \frac{27}{4} \right]
= \frac{54\pi}{4}
= \frac{27\pi}{2}
\]
The exact volume of the solid is \( \frac{27\pi}{2} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2e676c86-958a-4381-909f-e29a5bb4ea55%2F69f2bf61-d4a7-4965-b83f-b9dc28f148e2%2Fp6hwfr_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the volume of the solid obtained by rotating the region bounded by the curves \( y = 3x - x^2 \) and \( y = 0 \) about the \( y \)-axis. Give an exact answer in terms of \( \pi \).
**Explanation:**
To solve this problem, use the method of cylindrical shells. First, identify the region bounded by the equations in the xy-plane. The curve \( y = 3x - x^2 \) is a downward-facing parabola with roots at \( x = 0 \) and \( x = 3 \). The line \( y = 0 \) is the x-axis.
The volume of the solid is given by the integral:
\[
V = 2\pi \int_{a}^{b} x \cdot f(x) \, dx
\]
Here, \( f(x) = 3x - x^2 \), and the limits of integration \( a \) and \( b \) are the x-values where the parabola intersects the x-axis, \( 0 \) and \( 3 \).
Calculate the integral:
\[
V = 2\pi \int_{0}^{3} x(3x - x^2) \, dx
= 2\pi \int_{0}^{3} (3x^2 - x^3) \, dx
= 2\pi \left[ x^3 - \frac{x^4}{4} \right]_{0}^{3}
\]
Evaluate the integral:
\[
= 2\pi \left[ (27 - \frac{81}{4}) - (0 - 0) \right]
= 2\pi \left[ \frac{108}{4} - \frac{81}{4} \right]
= 2\pi \left[ \frac{27}{4} \right]
= \frac{54\pi}{4}
= \frac{27\pi}{2}
\]
The exact volume of the solid is \( \frac{27\pi}{2} \).
Expert Solution
Step 1: Finding intersection points
To find the volume of the solid obtained by rotating the region bounded by the curves
The volume
Where
First, let's find the intersection points:
This gives us
Step by step
Solved in 3 steps
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