Does anyone know how to do #105?
#Triple integral with a transformation and then spherical coordinates
thorny roost
shy sentinelBOT
thorny roost
teacher says the answer is 64pi/105 but i have no idea how he got there
thin iglooBOT
bacc (unhelpful)
thorny roost
yes
idle kettle
jacobi
crimson berry
Does anyone know how to do #105?
oh so youve already done the jacobian?
i was typing out an explanation of that but no worries if so : P
idle kettle
oh mb
thorny roost
yeah it's 144
crimson berry
alright, so now you just need the bounds for the integral then right?
thorny roost
its just, when you sub in, does the integral become 9u^2 * 36v^2 * 64z^2 ?
crimson berry
i think you meant for that z at the end to be a w, and if you did then yes exactly!
thorny roost
oh yeah w
crimson berry
because the integrand is just x^2 y^2 z^2 so that's just the product of our subs squared
thorny roost
okay cuz when i see problems online they just put in the jacobian and then say the integral is uvw and it made me incredibly confused
idle kettle
its just, when you sub in, does the integral become 9u^2 * 36v^2 * 64z^2 ?
it shouldnt be this
thorny roost
why not?
crimson berry
oh am i wrong?
thorny roost
idk im confused
idle kettle
The substitution is supposed to remove the factors
thorny roost
but isn't x = 3u so when u sub u in for x you get (3u)^2?
idle kettle
so you can apply spherical coords
thorny roost
IT DOES???
idle kettle
but isn't x = 3u so when u sub u in for x you get (3u)^2?
but look at the denominator
you divide by 9
9/9 = 1
and so on
idle kettle
oh bruh i am stupod
crimson berry
nahh no worries lol you almost gave me a heart attack!
idle kettle
the integrand is a product of these
thorny roost
im in the middle of a heart attack
crimson berry
LOL we're good now
thorny roost
none of my answers match the answer key 😭
crimson berry
so now that we have an integral in terms of u v and w, we just need to figure out the bounds right?
thorny roost
well you swap to spherical coords
crimson berry
yeah exactly
thorny roost
cuz the region is a sphere
crimson berry
so we do another "jacobian" but we really know what it is by heart now
thorny roost
and the bounds are 0 -> 2pi, 0 -> pi and 0->1
yup row^2 sin(phi) dpdphidtheta
and then with the integral in it with all the squares
times the 144
crimson berry
yeah, how do the squares simplify? so we converted our differentials to rho phi and theta, which means our squares will change too right?
(sorry im legit asking cause i dont have paper and my internet is really slow : P
idle kettle
,calc 144 * 3^2 * 6^2 * 8^2
thin iglooBOT
Result:
2.985984e+6
that's where i thought it was 9u^2 * 36v^2 * 64w^2 so itd change to 9(rho * sin(phi) * cos(theta))^2 * 36 (rho * sin(phi) * sin(theta))^2 * 64(rho * cos(phi))^2
crimson berry
the last one should be cos phi right?
idle kettle
what was the jacobian again
thorny roost
144
idle kettle
of spherical coords
crimson berry
looks good then! so we have 9 * 36 * 64 * rho^6 * sin ^4 (phi) * cos (theta) ^2 * sin(theta) ^ 2 from the integrand, right?
times the jacboain we would then have 9 * 36 * 64 * rho^8 * sin ^5 (phi) * cos (theta) ^2 * sin(theta) ^ 2
thorny roost
yes
crimson berry
alright cool! oh and times the 144 from earlier, i forgot that
thorny roost
yup
idle kettle
,w 144 * Integrate[9(rho * sin(phi) * cos(theta))^2 * 36 (rho * sin(phi) * sin(theta))^2 * 64(rho * cos(phi))^2 * rho^2 sin(phi), {phi,0,pi}, {theta,0,2pi}, {rho,0,1}]
thin iglooBOT
crimson berry
alright so now we have everything ready to go, did you have trouble integrating any of these variables in particular?
or was everything fine you just got a different answer? it seems like you get the idea, so it may be a small mistake somewhere in the math?
thorny roost
honestly i think the teacher might just be wrong
thorny roost
but he's been doing random stuff for the entire last five problems of the answer key
so i think he's just forgetting to actually sub in the (3u) instead of u
thorny roost
in any case it seems like you understand the process
ty 😭
thorny roost
,w 144 * Integrate[9(rho * sin(phi) * cos(theta))^2 * 36 (rho * sin(phi) * sin(t...
ty 🫡
idle kettle
did you get this btw?
thorny roost
yeah something around that
idle kettle
,w Integrate[x^2y^2z^2, {x,-3,3}, {y, -6sqrt(1-x^2/9), 6sqrt(1-x^2/9)}, {z, -8sqrt(1-x^2/9-y^2/36), 8sqrt(1-x^2/9-y^2/36)}]
thin iglooBOT
verbal ocean
Could someone help me?😭
idle kettle
dxdydz maybe try that too
,w Integrate[x^2y^2z^2, {z,-8,8}, {y, -6sqrt(1-z^2/64), 6sqrt(1-z^2/64)}, {x, -3sqrt(1-z^2/64-y^2/36), 3sqrt(1-z^2/64-y^2/36)}]
thin iglooBOT
idle kettle
yea you can show this