Question

Find the centroid of the area above$\mathit{y}\mathbf{=}{\mathit{x}}^{\mathbf{2}}$and below y = c (c > 0).

Short answer

The centroid$\left(\overline{x},\overline{y}\right)=\left(\frac{3}{8}{c}^{\frac{1}{2}},\frac{3}{5}c\right)$

Step-by-step solution

Given values

The values of the centroid is as follows:

$y=x^2$,$y=c\left(c>0\right)$

Concept of area of centroid

When finding the centroid of a two-dimensional shape, looking for both an x and a y coordinate, represented as x̄ and ȳ respectively. Collectively, this x and y coordinate is the centroid of the shape.$\mathit{c}\mathbf{=}\left(\overline{x,}\overline{y}\right)$

Find the area

The value of c must be$c⩾0$.

Find the area of centroid using the formula,

$A=\underset{0}{\overset{\sqrt{x}}{\int }}\text{d}x\underset{x^2}{\overset{c}{\int }}\text{d}y=\underset{0}{\overset{\sqrt{c}}{\int }}\text{d}x\left(c-x^2\right)$

role="math" localid="1664232091282" $=\left[cx-\frac{1}{3}{x}^3\right]|0^{\sqrt{c}}$

$=\frac{2}{3}{c}^{\frac{3}{2}}$