Find the centroid of region R

Question

Let R be the region below the curve y=4-x², to the left of y=3x and above the x-axis. Find the centroid of the region R

Answer 1

You can find the centroid (x_c,y_x) by finding the double integral from -4 to 1 in x and 3x to 4-x² in y of the expression xdydx and ydydx each divided by the Area of the region.

The area of the region is obtained from the integral from points of intersection of (y₂-y₁)dx or the same double integral as above of dydx. The intersection points are found by equating y₁ and y₂ : 4-x² = 3x and factoring to find the 0s of the quadratic. (-4 and 1)

I get the area = 125/6

I did the single integral of xΔydx and got -125/4. dividing by area, I obtained x_c = -3/2

The double integral of ydydx = -125/3 and the y_C = -2

I don't guarantee the results.

I recommend symbolab.com double integral calculator. It shows steps as well, so you can check your math.

You can also find the centroid using single integrals for y using yΔx(y)dy but you have to split up the integral and express x as a function of y and deal with sqrts.