Use cylindrical coordinates. Find the volume of the solid that lies within both the cylinder x2 + y2 = 25 and the sphere x2 + y2 + z2 = 49.

Answer:Step-by-step explanation:we are asked to find the volume of solid that lies within both the cylinder [tex]x^2 + y^2 = 25[/tex]and the sphere[tex]x^2 + y^2 + z^2 = 49.[/tex]Conversion from rectangular to cylindrical is[tex]x=rcost\\y = rsint\\z=z[/tex]|J| =rIn cylindrical coordinates the volume is bounded by the cylinder r=5 and[tex]r^2+z^2 =49[/tex]Hence we can write volume as[tex]\int \int \int dxdydz\\=\\\int _0^5 \int_0^{2\pi} \int_{-\sqrt{49-r^2} } ^{\sqrt{49-r^2} rdzdtdr\\= 2\pi \int _0^5 (2\sqrt{49-r^2} rdr\\=4\pi (-(49-r^2) (2/3)\\= \frac{4\pi}{3} (343-48\sqrt{6} )[/tex]