(2). ( D=left(x, y, z) in mathbbR^3: x, y, z geq 0, x+2 y+3...

Transcript

00:01 So here we have the region d, which is a set of points x, y, and z, such that x, y, and z are all greater than or equal to 0, and x plus 2y plus 3z is less than or equal to 6.

00:15 We want to go ahead and compute the triple integral of the region d of y, dx, dy, and dz.

00:24 All right, so let's first go ahead and consider our region.

00:32 So what we want to do is go ahead and solve for z in this first inequality.

00:40 So we're going to have 3z is less than or equal to 6 minus x minus 2y.

00:47 So z is then going to be less than or equal to 6 minus x minus 2y over 3.

00:53 So that means our bounds of z are going to go from 0 to 6 minus x minus 2y over 3.

01:00 And again this follows because we only have positive z and then we have a plane that's in the first quadrant or first octant excuse me so we're just looking at the area below that okay so now what we need to do is determine the limits for y so what we have is x plus 2y is less than or equal to 6 which means that when we solve for y we get y is less than or equal to 6 minus x over 2.

01:33 So y then goes from 0 to 6 minus x over 2.

01:38 And then what we have here is the maximum of x can be 6, so x is going to go from 0 to 6.

01:46 So setting up our triple integral, we're going to have 0 to 6, 0 to 6 minus x over 2, and then 0 to 6 minus x minus 2y over 3 and then y dz dy dx.

02:06 So we now need to go ahead and integrate with respect to z first.

02:11 So when we integrate with respect to z, we're going to have z times y from 0 to 6 minus x minus 2y over 3 and that's going to then be equivalent to y over 3 times 6 minus x minus 2y.

02:29 So this is then going to be 1 third integral 0 to 6 minus x over 2 of 6y minus xy minus 2y squared dy.

02:41 Y.

02:43 So this equals 1 third multiplied by 3 y squared minus x over 2 y squared minus 2 thirds y cubed.

02:54 And we go from 0 to 6 minus x over 2.

02:59 So then this is equivalent to 1 third times 3 times 6 minus x over 2 squared minus x over 2 times 6 minus x over 2 squared.

03:13 And then minus 2 thirds times 6 minus x over 2 cubed...

Question

(2). ( D=left{(x, y, z) in mathbb{R}^{3}: x, y, z geq 0, x+2 y+3 z leq 6 ight} ), [ iiint_{D} y d x d y d z ]

Question

(2). ( D=left{(x, y, z) in mathbb{R}^{3}: x, y, z geq 0, x+2 y+3 z leq 6 ight} ), [ iiint_{D} y d x d y d z ]

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