Determine the coordinates of the centriod of the region bounded by the graphs f x =4 -x^2 and g(x) =x + 2

Question

RizerMix · Accepted Answer

We want to find the coordinates of the centroid of the region bounded by the graphs

f (x) = 4 - x^{2}

and

g (x) = x + 2

.
To find the centroid, we need to find the values of

x

and

y

that satisfy the following equations:

\bar{x} = \frac{\int_{a}^{b} x \cdot f (x) d x}{\int_{a}^{b} f (x) d x} and \bar{y} = \frac{\int_{a}^{b} g (x) d x}{b - a}

where

\bar{x}

and

\bar{y}

are the

x

-coordinate and

y

-coordinate of the centroid, respectively, and

a

and

b

are the

x

-values where the two curves intersect.
First, let's find the intersection points by setting

f (x) = g (x)

:

4 - x^{2} = x + 2

Solving for

x

, we get:

x^{2} + x - 2 = 0

Factoring, we get:

(x + 2) (x - 1) = 0

So the two curves intersect at

x = - 2

and

x = 1

.
Next, we can find the integrals needed for

\bar{x}

and

\bar{y}

.

\bar{x} & = \frac{\int_{- 2}^{1} x \cdot f (x) d x}{\int_{- 2}^{1} f (x) d x}

= \frac{\int_{- 2}^{1} x (4 - x^{2}) d x}{\int_{- 2}^{1} (4 - x^{2}) d x}

= \frac{{[\frac{1}{2} (4 x^{2} - \frac{1}{4} x^{4})]}_{- 2}^{1}}{{[4 x - \frac{1}{3} x^{3}]}_{- 2}^{1}}

= \frac{\frac{9}{8}}{\frac{55}{3}}

= \frac{27}{440}

\bar{y} = \frac{\int_{- 2}^{1} g (x) d x}{1 - (- 2)}

= \frac{\int_{- 2}^{1} (x + 2) d x}{3}

= \frac{{[\frac{1}{2} x^{2} + 2 x]}_{- 2}^{1}}{3}

= \frac{9}{6}

= \frac{3}{2}

Therefore, the coordinates of the centroid are

(\bar{x}, \bar{y}) = (\frac{27}{440}, \frac{3}{2})

.

Determine the coordinates of the centriod of the