15.4: Applications of Double Integrals

Objective:
6. Applications with double & triple integrals - center of mass

[@arsalanmalik3801]

This video singlehandedly helped me get 20% of my entire EM course and I passed. Love you and thank you 🖤

[@Mary-mj7le]

Thanks! My professor in covid has never posted a lecture so this helps a lot... She just has "office hours"

[@eiao99]

Good explanation for the book's concepts, thanks for sharing!

[@yaredyirgalem3153]

You have no idea how much ur helping me out Tnx alot from 🇪🇹 ethiopia

[@ahihinah9726]

excuse , on the #3 , the center of mass on x is 0 because they are already balance on x-axis since (-a,0) and (a,0) right ? ( hope you reply my question , you are a great teacher )

[@alexandraniedden5337]

Yes, x = 0 because the figure is symmetric over the y-axis. It's more than the fact that the points are (-a,0) and (a,0) - the entire figure needs to be symmetric with respect to x, not just those two points. I hope that makes sense.

[@ahihinah9726]

@@alexandraniedden5337 i got it , thank you very much

[@enriquesoler3150]

in 23:59, when doing the double integral for the y coordinate of center of mass, where is r*sin(theta) coming from?

[@alexandraniedden5337]

r*sin(theta) is the polar conversion for y (when converting from rectangular to polar form). r*cos(theta) is used to convert x from rectangular to polar.

[@adrianramirez8832]

Shouldn't y bar for the last example be 3a/4pi? How did it simplify to 2pi when the only thing that gets crossed out is the a^3? (a^4)/4 * 3/a^3pi = 3a/4pi

[@alexandraniedden5337]

The sin(theta) integral gives us a two (-cos(pi) + cos(0) = 2), so we end up with (a^4/4)*2 = (a^2/2). The answer is correct as given.

[@beastvenom4193]

I would love to learn entire maths from you😁😁

[@juliak9899]

Hi, at around 18:48, you say that we don't need to calculate both x bar and y bar for example 1 because they will be the same. Why is that and how could I know for future examples when to only check for one (i.e. how can I know for future problems if x bar and y bar will give the same result)?

[@alexandraniedden5337]

Because the figure is symmetric with respect to the line y = x, we know that the center of mass will have the same values for the x- and y-coordinates. This should make sense looking at the figure - if we had to approximate the center of mass, we can guess that the x- and y-coordinates will be the same.

[@juliak9899]

@@alexandraniedden5337 Ohh, yes you're right, I didn't catch that, thank you!

[@ngawangr5324]

shawty said sigma

[@solomunallouche8425]

in the last example why is the x bar = 0 ?

[@rosemariez.seraga602]

Hello how did u get the mass 1/24? Need solution please

[@benjaminbenjamin8834]

Please tell which book you were using for these lecture videos?

[@aadityasuthar2199]

Thomas Calculus

[@wisdomwayeducation]

THANKS SIS VERY MUCH

[@abdulhadifayad4671]

Sorry to bother you Ms. but at 21:20 how did you find the equation of density?

[@alexandraniedden5337]

Questions are never (or rarely) a bother! The first sentence of the problem helped me write the equation for density. We know that density is proportional to distance from the center. Center is (0, 0), so distance from center is d = sqrt[(x - 0)^2 + (y - 0)^2] which simplifies to sqrt(x^2 + y^2). For the proportional part then, we get density = k*sqrt(x^2 + y^2). From the equation, as density increases, so does the distance (hence proportional).

[@geraldcalero4179]

At 14:32 how do you find the eq y=2-2x?

[@alexandraniedden5337]

I started with slope = (change in y) / (change in x). In this case to go from the top point to the bottom point, y changes by 2 and x changes by 1, with a downward slope, so m = -2. The y-intercept is 2 (the point (0, 2)). I then used slope-intercept form: y = mx+b, so y = -2x+2.

[@light-qn2jb]

thank u