Find the average of x2 y2 over the triangle with vertices

Question

If f is a continuous function of two variables x ∈ R and y ∈ R, defined over some region D ∈R², then the average value of f on D is given by

f_av = 1/A ∫∫_D f(x,y) dA,

where A is the area of the region D.

Use this definition of fav to answer the following questions.

Find the average value of y sin (xy) over the rectangle

R = {(x,y) : 0 ≤ x ≤ 1 ,0 ≤ y ≤ y ≤ π/2}

find the average of x² + y² over the triangle with vertices (0, 0), (1, 0), and (1, 1)

c. If the centroid of a region d is given by (x‾, y‾), where x‾ is the average value of x over d and y‾ is the average value of y over D, find the coordinates of the centroid of the region bounded by the curves y = 2x² and y=12 - x².