Find the center of mass of the homogenous lamina in the first quadrant bounded by
y = 0,

x = 0,

f(y) = −y2 + 81.

To find the center of mass of a homogeneous lamina in the first quadrant bounded by the curves \( y = 0 \), \( x = 0 \), and \( x = -y^2 + 81 \), we need to determine the coordinates of the centroid, \( (\bar{x}, \bar{y}) \).

Since the lamina is homogeneous, the density is constant and cancels out in the calculations. The centroid (center of mass) is calculated using the following formulas for \( \bar{x} \) and \( \bar{y} \):

\[
\bar{x} = \frac{1}{A} \int \int x \, dA
\]
\[
\bar{y} = \frac{1}{A} \int \int y \, dA
\]
where \( A \) is the area of the lamina, and \( dA \) represents the differential element of area.

### Step 1: Find the Area \( A \)
The area of the region is given by:
\[
A = \int_{y=0}^{y=9} \left( -y^2 + 81 \right) \, dy
\]
This integral bounds \( y \) from 0 to 9 because the curve intersects the x-axis when \( y = 9 \) (solving \( -y^2 + 81 = 0 \)).

### Step 2: Calculate \( \bar{x} \)
\[
\bar{x} = \frac{1}{A} \int_{y=0}^{y=9} \left( -y^2 + 81 \right) \cdot \left( -y^2 + 81 \right) \, dy
\]

### Step 3: Calculate \( \bar{y} \)
\[
\bar{y} = \frac{1}{A} \int_{y=0}^{y=9} \frac{1}{2} \left( (-y^2 + 81)^2 - (-y^2)^2 \right) \

Learn how to find the center of mass of a homogeneous lamina in the first quadrant bounded by y = 0, x = 0, and x = -y^2 + 81. This problem involves calculating the centroid using calculus and coordinate geometry techniques. Suitable for college-level mathematics.

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