Find the volume with integral of object bounded by z = 0, z = x*y, z + x + y = 1; I need to find the volume of z = 0; z+x+y = 1; z = x*y.The object. But I cannot make limits for the integral.I can make this integral The integral. But it will mean that we find this piece of objectWhole object that is needed to substract. So can you explain to me how to find this piece of object and then substract Piece of object or make a new one integral for whole this object. The answer must be 17/12-2ln(2) = 0.03

Consider for a fixed value of $x$ which ca n varies from $x = 0$ to $x = 1$. In between $0$ and $1$, the courses form a triangle with height and base length calculated as below.

$$z = xy$$ $$y + z = 1 - x$$

Solving this equations we have

$$y = \frac{1 - x}{1 + x}$$ $$z = \frac{x(1 - x)}{1 + x}$$

The height of the above triangle is given by the $y$ value above and the base is the y intercept given by $(1 - x)$.

Hence the volume is given by

$$\int_{0}^{1} \frac{1}{2}\frac{x(1 - x)}{1 + x}(1 - x) dx = \frac{17}{12} - 2ln2$$