Question

Calculate the integral of f(x,y)=8x over the region  bounded above by y=x(2−x) and below by x=y(2−y).

Answer 1

`(1)/(3)`

Explanation:

Let's first find the intersection points.

`y = x(2-x)\\x = y(2-y)\\y = y(2-y)(2-y(2-y))\\x = x(2-x)(2-x(2-x))`

So intersection points for x = 0, 1 and for y = 0, 1.

Now, let's write x = y(2-y) instead of y.

`x - 1 = 2y - y^2 - 1\\x - 1 = -(y-1)^2\\1-x = (y-1)^2\\y = 1 - √(1-x)`

`\int_D f(x,y)dA=\int\limits^1_0[x(2-x)-(1-√(1-x))]dx=\\\\= (x^2-(x^3)/(3) -x-(2)/(3)(1-x)^{(3)/(2) } )|^1_0=-(1)/(3) +(2)/(3)= (1)/(3)`