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.

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.

in progress 0
Luna 3 days 2021-10-12T07:08:38+00:00 1 Answer 0

Answers ( )

    0
    2021-10-12T07:09:44+00:00

    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 spherex^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} )

Leave an answer

Browse
Browse

27:3+15-4x7+3-1=? ( )