Use cylindrical coordinates to find the total mass and the z...

Use cylindrical coordinates to find the total mass and the z-coordinate of the center of mass of an object occupying the region inside the solid cylinder x2 + y2 ≤ 4 with 0 ≤ z ≤ 6 if the mass density is 𝜎(x, y, z) = ze−x2 − y2 kg/m3. M = kg z =

Transcript

0:00 Hello everyone.

00:02 In the given question we have to find the center of mass of the cylinder of radius 2 and height 4 and the mass density is e to the power minus jad, right, where j is the height.

00:12 So we have to find the center of mass.

00:15 So first we'll find, we'll write the cylindrical coordinates, that is x is equals to r cost theta, y is equals to our sine theta, and jad, which is given as jad.

00:27 These are the cylindrical coordinates.

00:29 And the volume dv will be will become r d theta d z right now we will define the region the region w is where r is from zero to two radius right the height is from zero to four and the theta value is from zero to two pi now we'll find the value of m xy so m xy is equal to we know from the definition that is jad delta function x y z dv so we'll substitute the limits the limits is from 0 to 2 pi radius is from 0 to 2 height is from 0 to 4 and j z value is jd multiplied by delta function that is a to the power minus jd multiplied by dv dv is r d theta d z d r d z d r so first we'll indicate with respect to see first we can rewrite this as we know that here in this we know that integration of x a minus a to the power j d z is equals to minus z equal minus eight to about this came from the integration by parts right so first we will integrate we can rewrite it as 0 to 2 pi d theta dot integration of 0 to 2 to 2 r multiplied by minus jad a to b minus z from 0 to 4 d r right we integrated with respect to j now we can write it as 2 pi multiplied by r squared by 2 from 0 to 2 right multiplied by minus 4 a to the bar minus 4 so this comes out to be 4 pi into 1 minus 5 a to the bar minus 4 now we have to find total mass so total mass am is equals to integration from 0 to 2 pi 0 to 2 0 to 4 r to 2 d to d d teta so when we integrate it we can rewrite it as d theta multiplied by integration from 0 to 2 r 1 minus 8 2 2 2 2 d r we integrated with respect to jett right we can simplify it as 2 pi into 2 1 minus 8 to the power minus 4...

Question

Use cylindrical coordinates to find the total mass and the z-coordinate of the center of mass of an object occupying the region inside the solid cylinder x2 + y2 ≤ 4 with 0 ≤ z ≤ 6 if the mass density is 𝜎(x, y, z) = ze−x2 − y2 kg/m3. M = kg z =

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