Evaluate the integral triple integral E 2xdV where E x y z

Evaluate the integral triple integral E 2xdV where E = {(x, y, z) | 0

Solution

_E 2x dv = _{[0 to 2] [0 to (4 -y²)] [0 to y]} 2x dz dx dy

= _{[0 to 2] [0 to (4 -y²)][0 to y]} 2x(z) dx dy

= _{[0 to 2] [0 to (4 -y²)]} 2xy dx dy

= _{[0 to 2][0 to (4 -y²)]} 2(x²/2)y dy

= _{[0 to 2]} ((4 -y²))²y dy

= _{[0 to 2]} 4y -y³ dy

= _{[0 to 2]} 4(y²/2) -y⁴4

= 2(2)² - 2⁴/4

= 8 - 4 = 4