Using Pappus' theorem to find the moment of a region

[pc2-brazil]

Homework Statement

R is the region limited by the semi-circumference $\sqrt{r^2 - x^2}$ 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 $\bar{y}$ 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:
$$V = 2\pi\bar{y}A$$
The moment of a plane region with respect to the x-axis is given by:
$$M_x = \bar{y}A$$
where $\bar{y}$ is the y-coordinate of the centroid of the region, or:
$$M_x = \int_a^b f(x)\times \frac{f(x)}{2} dx$$
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 $\frac{f(x)}{2}$ 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 $y = \sqrt{r^2 - x^2}$ and the x-axis is $\bar{y} = \frac{4r}{3\pi}$. So, the vertical coordinate of the centroid with respect to the line y = -4 would be $\bar{y} = \frac{4r}{3\pi}+4$. Since the area A is equal to $\frac{\pi r^2}{2}$, the moment should be obtained by:
$$M = \bar{y}A = \left ( \frac{4r}{3\pi} + 4\right )\frac{\pi r^2}{2}$$
I also get this same value if I calculate it by the following integral, which comes from the definition of moment of a region:
$$M = \int_{-r}^r \sqrt{r^2-x^2} \left (\frac{\sqrt{r^2-x^2}}{2} + 4 \right ) dx$$
But this doesn't lead to the correct answer, which is:
$$\frac{1}{2}r^3\left (\pi+\frac{4}{3}\right )$$

Thank you in advance.

 

[mighty2000]

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!