Volumes of revolution by the theorem of Pappus: area times the distance the centroid travels. 60 problems with worked solutions; in 60 of them every
equation is proved by a computer algebra system.
4 worked problems
Try each one before opening the solution. Every line is labeled with what checked it. How we verify
The region under \( \displaystyle y = 1x \) for \( \displaystyle 0 \le x \le 1 \) is rotated about the \( \displaystyle x \)-axis. Use the theorem of Pappus to find the volume.
Pappus: V = A · 2πd, where d is the distance from the centroid to the axis.
✓ Nihil obstat Lines: 3 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
The disk \( \displaystyle (x - 4)^2 + y^2 \le 1 \) is rotated about the \( \displaystyle y \)-axis. Use the theorem of Pappus to find the volume of the torus.
Pappus: V = A · 2πd, where d is the distance from the centroid to the axis.
Reviewed
\[ 1^{2} \pi = \pi \]
The disk's area, πa².✓ Proved
The centroid is the center (4, 0), 4 from the axis.
Reviewed
\[ 2 \pi 4 \pi = 8 \pi^{2} \]
V = 2π·d·A.✓ Proved
Answer \( 8 \pi^{2} \)
✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
The triangle with vertices \( \displaystyle (0, 0) \), \( \displaystyle (4, 0) \) and \( \displaystyle (0, 3) \) is rotated about the \( \displaystyle x \)-axis. Use the theorem of Pappus to find the volume.
Pappus: V = A · 2πd, where d is the distance from the centroid to the axis.
Reviewed
\[ 1 \cdot \frac{1}{2} \cdot 4 \cdot 3 = 6 \]
Area ½·base·height.✓ Proved
A triangle's centroid is the average of its vertices: ȳ = (0 + 0 + 3)/3 = 1.
Reviewed
\[ 2 \pi 1 \cdot 6 = 12 \pi \]
V = 2π·d·A.✓ Proved
Answer \( 12 \pi \)
✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
The rectangle \( \displaystyle 2 \le x \le 4 \), \( \displaystyle 0 \le y \le 1 \) is rotated about the \( \displaystyle y \)-axis. Use the theorem of Pappus to find the volume.
Pappus: V = A · 2πd, where d is the distance from the centroid to the axis.
Reviewed
\[ 2 \cdot 1 = 2 \]
Area = width × height.✓ Proved
The centroid is the center, at x = 3.
Reviewed
\[ 2 \pi 3 \cdot 2 = 12 \pi \]
V = 2π·d·A.✓ Proved
Answer \( 12 \pi \)
✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify