find the centroid of the area by parabolas x=y^2 and x^2=8y

Question

centroid of parabolas x=y^2 and x^2=8y

Answer 1

The equations of the parabolas are .

The centroid of the region has coordinates . It can be found using , where is the coordinates of the centroid of the differential element of area dA.

Use differential elements consisting of rectangular vertical slices of width dx and height y. This means that variable x will be the variable of integration.

In this case, and .

Find the intervals :

Equation 1 : .

Equation 2 : .

Solve equation 1 and 2.

.

.

First find the area.

.

Now observe that, .

Therefore,

and

.

Find the coordinates of the centroid.

     and     .

Centroid point : .