✒️QUADRANT
[tex]\large\bold\green{Formula:}[/tex]
Centroid by integration:
- [tex]\mathsf{A{\bar{x}}={\displaystyle\int}x\:dA}[/tex]
- [tex]\mathsf{A{\bar{y}}=\dfrac{1}{2}{\displaystyle\int}y\:dA}[/tex]
Refer to the figure
- [tex]\mathsf{y^2=4ax}[/tex]
- [tex]\mathsf{y=\sqrt{4ax}}[/tex]
[tex]\\[/tex]
Solving for A,
- [tex]\mathsf{dA=ydx}[/tex]
- [tex]\mathsf{A={\displaystyle\int_{0}^{a}}y\:dx}[/tex]
- [tex]\mathsf{A={\displaystyle\int_{0}^{a}}\sqrt{4ax}\:dx}[/tex]
- [tex]\mathsf{A=\sqrt{4a}{\displaystyle\int_{0}^{a}}\sqrt{x}\:dx}[/tex]
- [tex]\mathsf{A=2\sqrt{a}\left[\dfrac{x^{\frac{3}{2}}}{\frac{3}{2}}\right]_{0}^{a}}[/tex]
- [tex]\mathsf{A=2\sqrt{a}\left(\dfrac{2}{3}\right)\left(a^{\frac{3}{2}}\right)}[/tex]
- [tex]\mathsf{A=\dfrac{4}{3}a^2}[/tex]
Solving for [tex]\mathsf{\bar{x}}[/tex],
- [tex]\mathsf{A{\bar{x}}={\displaystyle\int}x\:dA}[/tex]
- [tex]\mathsf{A{\bar{x}}={\displaystyle\int_{0}^{a}}x(\sqrt{4ax})dx}[/tex]
- [tex]\mathsf{A{\bar{x}}=2\sqrt{a}{\displaystyle\int_{0}^{a}}x^{\frac{3}{2}}dx}[/tex]
- [tex]\mathsf{A{\bar{x}}=2\sqrt{a}\left[\dfrac{x^{\frac{5}{2}}}{\frac{5}{2}}\right]_{0}^{a}}[/tex]
- [tex]\mathsf{A{\bar{x}}=2\sqrt{a}\left(\dfrac{2}{5}\right)(a^{\frac{5}{2}})}[/tex]
- [tex]\mathsf{A{\bar{x}}=\dfrac{4}{5}a^3}[/tex]
- [tex]\mathsf{\bar{x}=\dfrac{\dfrac{4}{5}a^3}{A}}[/tex]
- [tex]\mathsf{\bar{x}=\dfrac{\dfrac{4}{5}a^3}{\dfrac{4}{3}a^2}}[/tex]
- [tex]\mathsf{\bar{x}=\dfrac{3}{5}a}[/tex]
Solving for [tex]\mathsf{\bar{y}}[/tex],
- [tex]\mathsf{A{\bar{y}}=\dfrac{1}{2}{\displaystyle\int}y\:dA}[/tex]
- [tex]\mathsf{A{\bar{y}}=\dfrac{1}{2}{\displaystyle\int_{0}^{a}}\sqrt{4ax}(\sqrt{4ax})dx}[/tex]
- [tex]\mathsf{A{\bar{y}}=\dfrac{1}{2}{\displaystyle\int_{0}^{a}}4ax\:dx}[/tex]
- [tex]\mathsf{A{\bar{y}}=2a{\displaystyle\int_{0}^{a}}x\:dx}[/tex]
- [tex]\mathsf{A{\bar{y}}=2a\left[\dfrac{x^2}{2}\right]_{0}^{a}}[/tex]
- [tex]\mathsf{A{\bar{y}}=a^3}[/tex]
- [tex]\mathsf{{\bar{y}}=\dfrac{a^3}{A}}[/tex]
- [tex]\mathsf{\bar{y}=\dfrac{a^3}{\dfrac{4}{3}a^2}}[/tex]
- [tex]\mathsf{\bar{y}=\dfrac{3}{4}a}[/tex]
Therefore, the answer is:
- [tex]\boxed{\mathsf{\left(\dfrac{3}{5}a,\:\dfrac{3}{4}a\right) \ or \ (0.6a, 0.75a)} }[/tex]
#CarryOnLearning
[tex]\qquad\qquad\qquad\qquad\qquad\qquad\tt{fri \: 03-04-2022} \\ \qquad\qquad\qquad\qquad\qquad\qquad\tt{3:02 \: pm}[/tex]
