2fc910…: “Thanks to a gray formula”

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Set-up integrals in rectangular, cylindrical, and spherical coordinates that will compute the volume of the top hemisphere (where $z$ is positive) of the sphere $x^2+y^2+z^2 = 49$ . $\int _{\answer {-7}}^{\answer {7}}\int _{\answer {-\sqrt {49-x^2}}}^{\answer {\sqrt {49-x^2}}}\int _{\answer {0}}^{\answer {\sqrt {49-x^2-y^2}}}\d z \d y \d x$ $\int _{0}^{\answer {2\pi }}\int _{\answer {0}}^{\answer {7}}\int _{\answer {0}}^{\answer {\sqrt {49-r^2}}} r\d z \d r \d \theta$ $\int _{0}^{\answer {2\pi }}\int _{0}^{\answer {\pi /2}}\int _{\answer {0}}^{\answer {7}} \rho ^2\sin (\varphi )\d \rho \d \varphi \d \theta$