let's move on to spherical coordinates:
"x = rcos\\varphi sin\\theta ,y = rsin\\varphi sin\\theta ,z = rcos\\theta"
"0 < r < 4,\\,\\,0 < \\varphi < 2\\pi ,\\,\\,0 < \\theta < \\pi"
Then
"{x^2} + {y^2} + {z^2} = {r^2}"
"\\begin{array}{l}\n\\int {\\int {\\int\\limits_E {({x^2} + {y^2} + {z^2})dV = \\int\\limits_0^{2\\pi } {d\\varphi } } } } \\int\\limits_0^4 {{r^2}dr} \\int\\limits_0^\\pi {{r^2}\\sin \\theta d\\theta } = \\int\\limits_0^{2\\pi } {d\\varphi } \\int\\limits_0^4 {{r^4}dr} \\int\\limits_0^\\pi {\\sin \\theta d\\theta } = \\\\\n = \\left. \\varphi \\right|_0^{2\\pi } \\cdot \\left. {\\frac{{{r^5}}}{5}} \\right|_0^4 \\cdot \\left. {( - \\cos \\theta )} \\right|_0^\\pi = 2\\pi \\cdot \\frac{{{4^5}}}{5}( - \\cos \\pi + \\cos 0) = \\frac{{2048}}{5}\\pi \\cdot (1 + 1) = \\frac{{4096\\pi }}{5}\n\\end{array}"
Answer: "\\frac{{4096\\pi }}{5}"
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