Answer
$$1$$
Work Step by Step
Since the region is symmetric with respect to both the x -and y- axes, we need to multiply the volume of one quadrant by $4$.
The bounds for $x$ are from 0 to 1 and the bounds for $y$ are from line $y=1-x$ to $1$.
$$Volume; V =\int_{0}^{1} \int_{1-x}^{1} x^2 \ dy \ dx =(4) \int_{0}^{1} \int_{1-x}^{1} x^2 \ dy \ dx \\=(4) \int_{0}^{1} (yx^2)_{1-x}^{1} x^2 \ dx \\= (4) \int_{0}^{1} (x^2-x^2+x^3) \ dx \\=(4) \int_{0}^{1} x^3 \ dx \\=(4) [\dfrac{x^4}{4}]_0^1\\=(4) (\dfrac{1}{4}-0) \\=1$$
