Chapter 15: Multiple Integrals - Section 15.2 - Double Integrals over General Regions - Exercises 15.2 - Page 883: 57

$\frac{4}{3}$

$y = x , y + x = 2$ $x= 0$ $y = x ......1$ $y+x = 2$ $y=2-x .....2$ Compare 1 and 2 $x=2-x$ $x=1$ $x=1,x=0$ $y=2-x,y=x$ $V = \int\limits_{0}^{1} \int\limits_{x}^{2-x}(x^2 +y^2) dydx$ $V = \int\limits_{0}^{1} \left(2x^2 -\frac{7x^3}{3} + \frac{(2-x)^3}{3}\right) dx$ $V=\frac{4}{3}$