Question

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 1

Answer:

Step-by-step explanation:

we are asked to find the volume of solid that lies within both the cylinder

$x^2 + y^2 = 25$

and

the sphere$x^2 + y^2 + z^2 = 49.$

Conversion from rectangular to cylindrical is

$x=rcost\\y = rsint\\z=z$

|J| =r

In cylindrical coordinates the volume is bounded by the cylinder r=5 and

$r^2+z^2 =49$

Hence we can write volume as

$\int \int \int dxdydz\\=\\\int _0^5 \int_0^{2\pi} \int_{-\sqrt{49-r^2} } ^{\sqrt{49-r^2} rdzdtdr\\= 2\pi \int _0^5 (2\sqrt{49-r^2} rdr\\=4\pi (-(49-r^2) (2/3)\\= \frac{4\pi}{3} (343-48\sqrt{6} )$