Pappus-Guldinus Theorems
Recommended Books on Amazon (affiliate links) | ||
---|---|---|
![]() |
![]() |
![]() |
Pappus-Guldinus Theorem 1
Here is a great video explaining the first theorem of Pappus-Guldinus.
video by Michel vanBiezen |
---|
Pappus-Guldinus Theorem 2
Here is a great video explaining the second theorem of Pappus-Guldinus.
video by Michel vanBiezen |
---|
Pappus-Guldinus Theorems Side-By-Side
Here is a great video explaining the theorems of Pappus-Guldinus side-by-side with an example. Here is the example. He has the same figure, a semi-circle, shown below. For the first part of the example, the semi-circle with radius \(R\) is a line which is rotated about the x-axis and he calculates the resulting surface area. For the second part of the example, the semi-circle defines an area which is also rotated about the x-axis. This results in a volume which he calculates. At the end, he compares the results.
Line of Rotation |
\( \bar{y} = 2R/\pi \) | |
---|---|---|
Area of Rotation |
\( \bar{y} = 4R/(3\pi) \) |
video by Michel vanBiezen |
---|
Okay, you should have all the information you need to solve these problems.
Practice
Unless otherwise instructed, solve these problems using the theorems on this page and give your answers in exact simplified form.
Use the theorems of Pappus-Guldinus to calculate the area produced when revolving the straight line from \((0,0)\) to \((6,3)\) about the x-axis.
Problem Statement |
---|
Use the theorems of Pappus-Guldinus to calculate the area produced when revolving the straight line from \((0,0)\) to \((6,3)\) about the x-axis.
Final Answer |
---|
\( A = 9\pi\sqrt{5} \approx 63.2 \)
We were not given units in the question, so none are required in the answer or you could say the answer is in square units.
Problem Statement
Use the theorems of Pappus-Guldinus to calculate the area produced when revolving the straight line from \((0,0)\) to \((6,3)\) about the x-axis.
Solution
video by Michel vanBiezen |
---|
Final Answer
\( A = 9\pi\sqrt{5} \approx 63.2 \)
We were not given units in the question, so none are required in the answer or you could say the answer is in square units.
Log in to rate this practice problem and to see it's current rating. |
---|
Use the theorems of Pappus-Guldinus to calculate the area produced when revolving the quarter circle with radius \(R\) shown in the figure about the x-axis.
Problem Statement |
---|
Use the theorems of Pappus-Guldinus to calculate the area produced when revolving the quarter circle with radius \(R\) shown in the figure about the x-axis.
Final Answer |
---|
\( A = 2\pi R^2 \) units2
Problem Statement
Use the theorems of Pappus-Guldinus to calculate the area produced when revolving the quarter circle with radius \(R\) shown in the figure about the x-axis.
Solution
video by Michel vanBiezen |
---|
Final Answer
\( A = 2\pi R^2 \) units2
Log in to rate this practice problem and to see it's current rating. |
---|
Use the theorems of Pappus-Guldinus to calculate the area produced when revolving the quarter circle with radius \(R\) shown in the figure about the x-axis.
Problem Statement |
---|
Use the theorems of Pappus-Guldinus to calculate the area produced when revolving the quarter circle with radius \(R\) shown in the figure about the x-axis.
Hint |
---|
From the previous problem, the center of mass is at the point \( (2R/\pi, 2R/\pi ) \)
Problem Statement |
---|
Use the theorems of Pappus-Guldinus to calculate the area produced when revolving the quarter circle with radius \(R\) shown in the figure about the x-axis.
Final Answer |
---|
\( A = \pi R^2( \pi - 2 ) \) units2
Problem Statement
Use the theorems of Pappus-Guldinus to calculate the area produced when revolving the quarter circle with radius \(R\) shown in the figure about the x-axis.
Hint
From the previous problem, the center of mass is at the point \( (2R/\pi, 2R/\pi ) \)
Solution
video by Michel vanBiezen |
---|
Final Answer
\( A = \pi R^2( \pi - 2 ) \) units2
Log in to rate this practice problem and to see it's current rating. |
---|
Use the theorems of Pappus-Guldinus to calculate the area produced when revolving circle with radius \(R = 2\) shown in the figure about the x-axis.   Units are in centimeters.
Problem Statement |
---|
Use the theorems of Pappus-Guldinus to calculate the area produced when revolving circle with radius \(R = 2\) shown in the figure about the x-axis.   Units are in centimeters.
Final Answer |
---|
\( A = 40\pi^2 \) cm2
Problem Statement
Use the theorems of Pappus-Guldinus to calculate the area produced when revolving circle with radius \(R = 2\) shown in the figure about the x-axis.   Units are in centimeters.
Solution
video by Michel vanBiezen |
---|
Final Answer
\( A = 40\pi^2 \) cm2
Log in to rate this practice problem and to see it's current rating. |
---|
Use the theorems of Pappus-Guldinus to calculate the area produced when revolving semi-circular line segment with radius \(R = 2\) shown in the figure about the x-axis.   Units are in centimeters.
Problem Statement
Use the theorems of Pappus-Guldinus to calculate the area produced when revolving semi-circular line segment with radius \(R = 2\) shown in the figure about the x-axis.   Units are in centimeters.
Solution
video by Michel vanBiezen |
---|
Log in to rate this practice problem and to see it's current rating. |
---|
Use the theorems of Pappus-Guldinus to calculate the area produced by revolving the straight line segment from the point \( (4,2) \) to the point \( (12,10) \) about the x-axis.
Problem Statement
Use the theorems of Pappus-Guldinus to calculate the area produced by revolving the straight line segment from the point \( (4,2) \) to the point \( (12,10) \) about the x-axis.
Solution
video by Michel vanBiezen |
---|
Log in to rate this practice problem and to see it's current rating. |
---|
Use the theorems of Pappus-Guldinus to calculate the volume produced by the circle with radius \(R = 3\)cm with center \( 10 \)cm above the x-axis when revolved about the x-axis.
Problem Statement
Use the theorems of Pappus-Guldinus to calculate the volume produced by the circle with radius \(R = 3\)cm with center \( 10 \)cm above the x-axis when revolved about the x-axis.
Solution
video by Michel vanBiezen |
---|
Log in to rate this practice problem and to see it's current rating. |
---|
Use the theorems of Pappus-Guldinus to calculate the volume produced when revolving the object shown in the figure about the x-axis.
Problem Statement
Use the theorems of Pappus-Guldinus to calculate the volume produced when revolving the object shown in the figure about the x-axis.
Solution
Log in to rate this practice problem and to see it's current rating. |
---|
Use the theorems of Pappus-Guldinus to calculate the volume produced when revolving the object shown in the figure about the x-axis.   The curved line is the equation \( y = 3x^2 \)
Problem Statement
Use the theorems of Pappus-Guldinus to calculate the volume produced when revolving the object shown in the figure about the x-axis.   The curved line is the equation \( y = 3x^2 \)
Solution
video by Michel vanBiezen |
---|
Log in to rate this practice problem and to see it's current rating. |
---|
Really UNDERSTAND Physics
Log in to rate this page and to see it's current rating.
To bookmark this page and practice problems, log in to your account or set up a free account.
Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.
| |
I recently started a Patreon account to help defray the expenses associated with this site. To keep this site free, please consider supporting me. |
---|
Support 17Calculus on Patreon |
|
---|