# Thread: center of mass

1. ## center of mass

Find the center of mass (x,y) of the region bounded by the graphs of f(x)=4-x^2 and g(x)=x+2.

2. Another hint: take a triangular piece of paper. From each corner, draw a line to the middle of the opposite side. Each line will intersect at a particular point. That point is your center of mass.

3. Hi

First draw the curves

Then compute the intersects of the 2 curves by solving $\displaystyle 4-x^2 = x+2$
Spoiler:

$\displaystyle (x-1)(x+2)=0 \implies x=-2 \r \:x=1$

The coordinates of the center of mass G are given by the formula

$\displaystyle x_G = \frac{\int\int x\:dx\:dy}{\int\int dx\:dy}$

$\displaystyle y_G = \frac{\int\int y\:dx\:dy}{\int\int dx\:dy}$

You can see that the area is defined by :
x between -2 and 1 ; y between (x+2) and 4-x² (see green line)

Therefore

Spoiler:

$\displaystyle x_G = \frac{\int_{-2}^{1}\int_{x+2}^{4-x^2} x\:dy\:dx}{\int_{-2}^{1}\int_{x+2}^{4-x^2} \:dy\:dx}$

$\displaystyle y_G = \frac{\int_{-2}^{1}\int_{x+2}^{4-x^2} y\:dy\:dx}{\int_{-2}^{1}\int_{x+2}^{4-x^2} \:dy\:dx}$

4. thanks but i dont understand double integers

5. Originally Posted by mikegar813
thanks but i dont understand double integers
You mean that you haven't studied double integrals ?

6. we have not learned double interals sorry