A lamina occupies the part of the rectangle 0x3 0y8 and the...

Name: a lamina occupies the part of the rectangle 0x3 0y8 and the density at each point is given by the function xy3x1y3 a what is the total mass b where is the center of mass 37755
Uploaded: 2023-08-19T01:52:30-08:00
Duration: 7 min 13 s
Channel: Vincenzo Zaccaro
Description: a lamina occupies the part of the rectangle 0x3 0y8 and the density at each point is given by the function xy3x1y3 a what is the total mass b where is the center of mass 37755

A lamina occupies the part of the rectangle 0<=𝑥<=3 , 0<=𝑦<=8 and the density at each point is given by the function 𝜌(𝑥,𝑦)=3𝑥+1𝑦+3 . A. What is the total mass? B. Where is the center of mass? ( , )

Transcript

00:01 Ok, so let's get started with part a of our exercise.

00:05 Well, here we need to compute the mass of our lamina.

00:11 Well, the mass is going to be an integral with respect to x from 0 to 2, an integral with respect to y from 0 to 5, of our density function 3x plus y plus 3, in dy and then in dx.

00:31 So what do we get? we get an integral from 0 to 2 with respect to x of 15x plus 25 halves, the integral of this one, plus 15 in dx.

00:50 Ok, now let's compute this integral with respect to x.

00:55 Easy.

00:56 This one is going to be 15 multiplied by 2 squared over 2, so 30, plus 25, the integral of this one, plus 30, the integral of this one.

01:11 So we get 85.

01:13 This is the mass of our lamina.

01:16 Now, let's compute the center of mass.

01:19 Well, the x coordinate is going to be an integral from 0 to 2, an integral from 0 to 5 with respect to y, of x multiplied by the density function.

01:33 So 3x squared plus xy plus 3x in dy and dx, and we need to multiply this integral by 1 over the mass.

01:52 So i'm going to write 1 over 85 here.

01:55 This is the x coordinate of the center of mass.

01:59 Ok, let me compute this integral.

02:02 Well, i'm going to rewrite this guy first.

02:06 So 1 over 85 multiplied by an integral from 0 to 2 of the integral of this one with respect to y, 15x squared, the integral of this one with respect to y, 25 halves x, the integral of this one with respect to y, 15x.

02:29 Perfect.

02:31 Ok, now let's compute this integral.

02:35 1 over 85 multiplied by the integral of this one with respect to x, well, 15 multiplied by 8 thirds.

02:46 The integral of this one with respect to x, this one is, well, 25, and the integral of this one with respect to x, ok, so this one is going to be 30.

03:00 Perfect.

03:02 Ok, so we get 1 over 85 multiplied by, ok, this guy here is going to be 120 over 3, so 40 plus 25 plus 30.

03:25 Ok, so we get 95 over 85.

03:30 Perfect.

03:30 Ok, now we need to compute the y coordinate of the center of mass...

Question

A lamina occupies the part of the rectangle 0<=𝑥<=3 , 0<=𝑦<=8 and the density at each point is given by the function 𝜌(𝑥,𝑦)=3𝑥+1𝑦+3 . A. What is the total mass? B. Where is the center of mass? ( , )

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