The centroid of the region enclosed by the graphs of and lies on the line

Question

The centroid of the region enclosed by the graphs of  and  lies on the line

Quizplus · Accepted Answer

To find the centroid of the region, we need to find the equations of the two curves and then find the coordinates of the centroid using calculus. 
The two given equations are $y=2x$ and $y=x^2$. 
To find the x-coordinate of the centroid, we can use the formula: 
$$\bar{x}=\frac{\int_{a}^{b}x\cdot f(x) dx}{\int_{a}^{b} f(x) dx}$$
where $$f(x) = (x^2 - 2x)$$
and $a$ and $b$ are the $x$-coordinates of the points of intersection of the two curves. The points of intersection can be found by setting the two equations equal to each other: 
$$2x = x^2 \quad \Rightarrow \quad x(x-2)=0 \quad \Rightarrow \quad x=0,2$$
So, the limits of integration are $a=0$ and $b=2$. Now, we can find the x-coordinate of the centroid:

$$\bar{x}=\frac{\int_{0}^{2} x(x^2-2x) dx}{\int_{0}^{2} (x^2-2x) dx}$$
$$\bar{x}=\frac{\frac{1}{4}(2)^4 - \frac{1}{3}(2)^3}{\frac{1}{3}(2)^3 - \frac{1}{2}(2)^2}$$
$$\bar{x}=\frac{8/3}{4/3}=\frac{2}{1}=2$$

To find the y-coordinate of the centroid, we can use the formula: 
$$\bar{y}=\frac{1}{A} \cdot \int_{a}^{b} f(x) dx$$
where A is the area of the region enclosed by the curves. A can be found by subtracting the smaller curve from the larger curve and integrating: 
$$A=\int_{0}^{2} (x^2-2x) - 2x dx$$
$$A=\int_{0}^{2} x^2-4x dx$$
$$A=\frac{1}{3}(2)^3 - 2(2)^2 + \frac{1}{3}(0)^3 - 2(0)^2 = \frac{8}{3}$$
Now, we can find the y-coordinate of the centroid:

$$\bar{y}=\frac{1}{8/3} \cdot \int_{0}^{2} x^2-2x dx$$
$$\bar{y}=\frac{3}{8} \cdot \left[\frac{1}{3}x^3-x^2ight]_{0}^{2}$$
$$\bar{y}=\frac{3}{8} \cdot \left[\left(\frac{8}{3}-4ight)-\left(0-0ight)ight] = \frac{1}{2}$$

Therefore, the coordinates of the centroid are $(2,\frac{1}{2})$. We can check that this point lies on the line $y=\frac{1}{2}x-1$ by substituting $x=2$ and $y=\frac{1}{2}$ into the equation: 
$$\frac{1}{2} = \frac{1}{2}(2) - 1.$$ That equation is true, so the answer is $\boxed{	extbf{(D)}}$.

The Centroid of the Region Enclosed by the Graphs of