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4 $begingroup$ I'm currently working on this question: Find the volume of the solid that lies within both the cylinder $x^2+y^2=1$ and the sphere $x^2+y^2+z^2=4$ . I decided to use polar coordinates so that the cylinder equation becomes $r^2=1$ and the sphere becomes $r^2+z^2=4$ . Solving for $z$ , I get the inequality $-sqrt{4-r^2}leq zleq sqrt{4-r^2}$ . Since I know what $r^2$ is, I plug that in to get the inequality where $z$ is between $-sqrt{3}$ and $sqrt{3}$ . Combining that to make a triple integral, I get: $int_0^{2pi}int_0^1int_{-sqrt{3}}^sqrt{3}rdzdrdtheta$ However, Slader has a different answer where they didn't plug in $sqrt{3}$ into the bounds. Why does plugging in the value for $r^2$ make the calculation wrong? Isn't $r^2$ always $1$ ? ...