Question

Calc III- Please help!
A lamina occupies the part of the disk x^2+y^2<=1 in the first quadrant. Find its center of mass if the density at any point is proportional to the square of the distance from the origin.

Answer 1

@dan815

Answer 2

@jim_thompson5910 @amistre64

Answer 3

@iambatman @aaronq

Answer 4

\[x ^{2}+y ^{2}\le1\]

Answer 5

$$
\Large \iint_D \rho (x,y) dA

$$

Answer 6

yup

Answer 7

but im having troubel setting up the density function

Answer 8

it would be easier to convert this problem to polar coordinates

Answer 9

ok so r=1

Answer 10

or r<=1

Answer 11

$$ \Large \rho (x,y) = k (\sqrt{ x^2 + y^2 })^2

$$

Answer 12

ahh i see

Answer 13

The region is described as the set of all \( (r, \theta ) \) such that \( 0\leq r\leq 1 \) and \( 0 \leq \theta \leq \pi/2 \).

Answer 14

\[\int\limits_{0}^{1}\int\limits_{0}^{2\pi}kr ^{3}d thetadr\]

Answer 15

wait the limit is pi/2 not 2pi

Answer 16

right, that will give you the mass . and then you need moment with respect to x axis and moment with respect to y axis .

Answer 17

its just that times x and y right?