Using Pappus' theorem to find the moment of a region

In summary, the conversation discusses using Pappus' theorem to find the moment of a region with respect to a given line. The formula for the moment of a region is explained, and the correct integral for this specific case is given. The correct answer is also provided as \frac{1}{2}r^3\left (\pi+\frac{4}{3}\right ).
  • #1
pc2-brazil
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3

Homework Statement


R is the region limited by the semi-circumference [itex]\sqrt{r^2 - x^2}[/itex] and the x-axis. Use Pappus' theorem to find the moment of R with respect to the line y = -4.

Homework Equations


Pappus' theorem:
If R is the region limited by the functions f(x) and g(x), then, if A is the area of R and [itex]\bar{y}[/itex] is the y-coordinate of the centroid of R, the volume V of the solid of revolution obtained by rotating R around the x-axis is given by:
[tex]V = 2\pi\bar{y}A[/tex]
The moment of a plane region with respect to the x-axis is given by:
[tex]M_x = \bar{y}A[/tex]
where [itex]\bar{y}[/itex] is the y-coordinate of the centroid of the region, or:
[tex]M_x = \int_a^b f(x)\times \frac{f(x)}{2} dx[/tex]
where a and b are the x-coordinates that limit the region R, f(x) is the height of the region at x (therefore, f(x)dx is the area of each element of area) and [itex]\frac{f(x)}{2}[/itex] is the centroid of the infinitesimal rectangle.
Moment of region is analogous to moment of mass; the center of mass is calculated by dividing moment of mass by the total mass; the centroid is calculated by dividing the moment of region by the total area of the region.

The Attempt at a Solution


The y-coordinate of the centroid of a semicircular area limited by [itex]y = \sqrt{r^2 - x^2}[/itex] and the x-axis is [itex]\bar{y} = \frac{4r}{3\pi}[/itex]. So, the vertical coordinate of the centroid with respect to the line y = -4 would be [itex]\bar{y} = \frac{4r}{3\pi}+4[/itex]. Since the area A is equal to [itex]\frac{\pi r^2}{2}[/itex], the moment should be obtained by:
[tex]M = \bar{y}A = \left ( \frac{4r}{3\pi} + 4\right )\frac{\pi r^2}{2}[/tex]
I also get this same value if I calculate it by the following integral, which comes from the definition of moment of a region:
[tex]M = \int_{-r}^r \sqrt{r^2-x^2} \left (\frac{\sqrt{r^2-x^2}}{2} + 4 \right ) dx[/tex]
But this doesn't lead to the correct answer, which is:
[tex]\frac{1}{2}r^3\left (\pi+\frac{4}{3}\right )[/tex]

Thank you in advance.
 
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  • #2


Hi there,

Thank you for your post. Your approach using Pappus' theorem and the formula for the moment of a region is correct. However, I believe your calculation for the moment is incorrect.

The moment of a region with respect to a line is given by the integral:
M = \int_a^b x \times f(x) dx
where a and b are the x-coordinates that limit the region R, and f(x) is the height of the region at x.

In this case, the integral should be:
M = \int_{-r}^r x \times \sqrt{r^2-x^2} dx

Solving this integral will give you the correct answer of \frac{1}{2}r^3\left (\pi+\frac{4}{3}\right ).

I hope this helps. Let me know if you have any other questions. Keep up the good work!
 

Related to Using Pappus' theorem to find the moment of a region

1. What is Pappus' theorem?

Pappus' theorem is a mathematical principle that relates the volume of a three-dimensional solid to the area of a two-dimensional cross-section and the distance traveled by that cross-section when rotated around an axis. It is often used to calculate the centroid, surface area, and volume of irregular shapes.

2. How is Pappus' theorem used to find the moment of a region?

Pappus' theorem can be used to find the moment of a region by first calculating the centroid of the region using the formula (x̄ = ∫x dA / ∫dA, ȳ = ∫y dA / ∫dA). Then, the moment of the region around a given axis can be found by multiplying the area of the region by the distance between the centroid and the axis.

3. What are the prerequisites for using Pappus' theorem to find the moment of a region?

In order to use Pappus' theorem to find the moment of a region, one must have a good understanding of calculus and geometric principles. Additionally, knowledge of the cross-sectional area and distance traveled by the cross-section when rotated around an axis are necessary to apply the theorem.

4. Can Pappus' theorem be used for all types of shapes?

Yes, Pappus' theorem can be used for any type of shape, including irregular and asymmetric shapes. As long as the cross-sectional area and distance traveled by the cross-section can be determined, the theorem can be applied.

5. Are there any limitations or drawbacks to using Pappus' theorem for finding the moment of a region?

One limitation of Pappus' theorem is that it only applies to solid objects with constant cross-sectional area. In addition, it can be difficult to determine the cross-sectional area and distance traveled for complex shapes, making the application of the theorem more challenging. It is also important to note that Pappus' theorem only applies to moments around a given axis, and not moments of inertia.

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