Find the centroid of the region bounded by the given curves.

Question

Quizplus · Accepted Answer

The region is bounded by the curves $y = x^2$, $x = 1$, and $y = 0$.

We can find the centroid $(\bar{x},\bar{y})$ using the following formulas:

$$\bar{x} = \frac{\int_{a}^{b} x f(x) dx}{\int_{a}^{b} f(x) dx}$$

$$\bar{y} = \frac{\int_{c}^{d} y g(y) dy}{\int_{c}^{d} g(y) dy}$$

where $f(x)$ is the distance of the curve from the $y$-axis, $g(y)$ is the distance of the curve from the $x$-axis, and $(a,b)$ and $(c,d)$ are the intervals of integration.

In this case, we have:

$$f(x) = x$$
$$g(y) = \sqrt{y}$$

and the limits of integration are:

$$a = 0, \quad b = 1, \quad c = 0, \quad d = 1$$

so we can calculate:

$$\bar{x} = \frac{\int_{0}^{1} x^3 dx}{\int_{0}^{1} x dx} = \frac{1}{4}$$

$$\bar{y} = \frac{\int_{0}^{1} y^{3/2} dy}{\int_{0}^{1} \sqrt{y} dy} = \frac{2}{5}$$

Therefore, the centroid is at $\left(\frac{1}{4}, \frac{2}{5}\right)$, which corresponds to choice (C).

Find the Centroid of the Region Bounded by the Given