Question

Sketch the region bounded by the curves y=x²/3, y=3, and x=3 and determine the centroid of the region.

Answer 1

The region bounded by the curves
`\(y = (x^2)/(3)\), \(y = 3\), and \(x = 3\)` is a triangular region. The centroid of this region is located at
`\((2, 2)\)`.

Step-by-step explanation:

To sketch the region, first, consider the curves
`\(y = (x^2)/(3)\) and \(y = 3\).` The intersection points occur where
`\((x^2)/(3) = 3\).`

Solving for
`\(x\),` we find
`\(x = 3\)` and
`\(x = -3\),` but since we're considering
`\(x \geq 0\),` the intersection point is
`\(x = 3\).`

The curves
`\(y = (x^2)/(3)\), \(y = 3\),` and
`\(x = 3\)` form a triangular region.

To find the centroid of a two-dimensional region, you need to determine the coordinates
`\((\bar{x}, \bar{y})\)` using the formulas
`\(\bar{x} = (1)/(A) \int x \, dA\)` and
`\(\bar{y} = (1)/(2A) \int (y_1 + y_2) \, dA\),` where
`\(A\)` is the area of the region.

In this case, the triangular region's centroid coordinates are
`\((\bar{x}, \bar{y}) = (2, 2)\)`.

This means that the center of mass of the triangular region is located at
`\((2, 2)\),` providing a balance point for the given geometry.

Answer 2

The centroid of the region bounded by the curves y=x²/3, y=3, and x=3 is located at the coordinates (2, 18/5).

Step-by-step explanation:

To determine the region's area and centroid, you need to find the points of intersection among the given curves. The curves y = x²/3 and y = 3 intersect at x = ±√9, which gives the bounds for integration. Integrating y=x²/3 from -3 to 3 provides the area bounded by the curves. The formula for the x-coordinate of the centroid,
`\( \bar{x} = (1)/(A) \int_(a)^(b) (1)/(2) (f(x))^2 \, dx \)`, where A is the area, gives the x-coordinate of the centroid.

Calculating this integral and dividing by the area provides the x-coordinate of the centroid. In this case, it's at x = 2. To find the y-coordinate of the centroid,
`\( \bar{y} = (1)/(A) \int_(a)^(b) f(x) \cdot (1)/(2) (f(x))^2 \, dx \)`, we use the same approach. After integration and division by the area, we get y = 18/5. Therefore, the centroid of the bounded region is at coordinates (2, 18/5).