Question

Sketch the region bounded by the curves y=4x-x² and y=x and determine the centroid of the region.

Answer 1

The centroid of the region bounded by the curves
`\(y=4x-x^2\) and \(y=x\) is \(\left((4)/(3), (2)/(3)\right)\).`

Step-by-step explanation:

The region bounded by the curves
`\(y=4x-x^2\) and \(y=x\)` can be found by setting the two equations equal to each other:
`\(4x-x^2 = x\)`. Solving this quadratic equation, we get the points of intersection as
`\(x=0\) and \(x=3\).`

To find the area between the curves, integrate the upper curve
`\(4x-x^2\)`minus the lower curve
`\(x\)` from
`\(x=0\) to \(x=3\)`. After finding the area, we need to calculate the
`\(x\)`-coordinate of the centroid using the formula
`\(\bar{x} = (1)/(A) \int_(a)^(b) x \cdot (f(x) - g(x)) \,dx\)`, where
`\(A\)` is the area,
`\(f(x)\)` is the upper curve, and
`\(g(x)\)` is the lower curve. Similarly, the
`\(y\)`-coordinate of the centroid is given by
`\(\bar{y} = (1)/(2A) \int_(a)^(b) [f(x)^2 - g(x)^2] \,dx\)`. Evaluating these integrals yields the centroid coordinates
`\(\left((4)/(3)`,
`(2)/(3)\right)\))`.