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2 $begingroup$ I want to calculate triple integral begin{equation}intlimits_{-1}^{1}intlimits_{-sqrt{1-x^2}}^{sqrt{1-x^2}}intlimits_{x^2+y^2}^1 2z dzdydx.end{equation} (the surface is $z=x^2+y^2$ , $0leq zleq 1$ .) In MAPLE, I have to calculate it, and the result is $$dfrac{2}{3}pi.$$ Now I want calculate the triple integral with cylindrical coordinates, become this begin{equation}intlimits_{0}^{2pi}intlimits_{0}^{1}intlimits_{r}^1 2zr dzdrdtheta.end{equation} begin{eqnarray} intlimits_{0}^{2pi}intlimits_{0}^{1}intlimits_{r}^1 2zr dzdrdtheta &=& intlimits_{0}^{2pi}intlimits_{0}^{1}left[z^2rright]_r^1drdtheta\ &=& intlimits_{0}^{2pi}intlimits_{0}^{1}left[r-r^3right]drdtheta\ &=& intlimits_{0}^{2pi}left[dfrac{1}{2}r^2-dfrac{1}{4}r^4right]_0^1dtheta\ &=& intlimits_{0}^{