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alffer1
Group Title
Multivariable integration help:
Compute the moment of inertia around the yaxis associated with the region y<x^2, x<2, assuming constant density delta = 5/32.
Not sure what to start with...
 10 months ago
 10 months ago
alffer1 Group Title
Multivariable integration help: Compute the moment of inertia around the yaxis associated with the region y<x^2, x<2, assuming constant density delta = 5/32. Not sure what to start with...
 10 months ago
 10 months ago

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oldrin.bataku Group TitleBest ResponseYou've already chosen the best response.1
the moment of inertia \((I_x,I_y)\) is computed using:$$I_x=\iint_R x^2 \delta(x,y)\ dx\ dy\\I_y=\iint_R y^2\delta(x,y)\ dx\ dy$$
 10 months ago

oldrin.bataku Group TitleBest ResponseYou've already chosen the best response.1
here \(R\) is given by \(x^2<y<x^2\) and \(2<x<2\) whereas \(\delta=5/32\) ergo:$$\begin{align*}I_x&=\int_{2}^2\int_{x^2}^{x^2}x^2\cdot\frac5{32}\ dy\ dx\\&=\frac5{32}\int_{2}^2 x^2\int_{x^2}^{x^2}\ dy\ dx\\&=\frac5{32}\int_{2}^2 2x^4\ dx\\&=\frac18\int_0^25x^4\ dx\\&=\frac18(2^50^5)\\&=4\end{align*}$$
 10 months ago

oldrin.bataku Group TitleBest ResponseYou've already chosen the best response.1
can you find \(I_y\)?
 10 months ago

alffer1 Group TitleBest ResponseYou've already chosen the best response.0
oh duh...I forgot I had 2 dimensions to work with. Thanks a ton!
 10 months ago

oldrin.bataku Group TitleBest ResponseYou've already chosen the best response.1
no problem :p it's the same integrand only \(y^2\)
 10 months ago

alffer1 Group TitleBest ResponseYou've already chosen the best response.0
wait so same as the first line in the bounds too?
 10 months ago

oldrin.bataku Group TitleBest ResponseYou've already chosen the best response.1
indeed
 10 months ago

alffer1 Group TitleBest ResponseYou've already chosen the best response.0
just a sec, somewhat confused, are the bounds changed to y^2? no, right?
 10 months ago

oldrin.bataku Group TitleBest ResponseYou've already chosen the best response.1
nope!
 10 months ago

alffer1 Group TitleBest ResponseYou've already chosen the best response.0
ok got the answers. you were great. Saved my neck here.
 10 months ago

oldrin.bataku Group TitleBest ResponseYou've already chosen the best response.1
$$I_y=\int_{2}^2\int_{x^2}^{x^2}y^2\cdot\frac5{32}\ dy\ dx=\frac5{32}\int_{2}^2\left[\frac13y^3\right]_{x^2}^{x^2}\ dx=\frac5{32}\int_{2}^2\frac23x^6dx$$so we get:$$I_y=\frac5{48}\int_{2}^2 x^6\ dx=\frac5{24}\int_0^2x^6\ dx=\frac5{24}\cdot\frac17(2^70^7)=\frac{5\cdot16}{3\cdot7}=\frac{80}{21}$$
 10 months ago
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