(1 point) Express the volume of the solid under the plane x...

(1 point) Express the volume of the solid under the plane x - 3y + z = 6 and above the region bounded by 2x + y = 4 and x² + y = 4. Volume = \int_a^b \int_c^d \text{________} \, dy \, dz where a = \text{________} b = \text{________} c = \text{________} d = \text{________} Find the volume. Volume = \text{________}

Transcript

00:01 Hi, in this video we're going to set up some double and triple integrals.

00:06 So we're going to find the volume of some regions, so let's start by kind of getting a sketch of the region.

00:13 So it's bounded by the plane z equals 2, which i probably won't draw the z axis, so we'll just stick to the x y because i think that'll give us all the information we need because the z boundary is just z equals 2, so that's sort of easy to visualize.

00:30 Y plane has more going on.

00:33 So it's bounded by the straight lines y equals x.

00:36 So we have this line, y equals x, and then x plus y equals 1, which you can think of as y equals 1 minus x.

00:45 So that's going to look something like if this is the point 1, then it's going to look like this.

00:51 And y equals 0, which is the x -axis.

00:55 So the region we're looking at is going to be this here.

01:04 So if we want to find the area of this, we have to integrate over this region.

01:12 So this point here where y equals x meets y equals 1 minus x, is when x equals 1 minus x, is when x equals 1 minus x, so x equals 1 1⁄2.

01:24 So this value is 1⁄2.

01:28 So when we integrate, we're going to integrate from over x first from 0 to 1⁄2, and then the y is going to go from zero to y equals x so until until x and then we're going to integrate two and do y d x so that was the first half of this triangle that was just this left half and now we're going to do the right half which is almost the same thing but now we integrate from one half from one half to one and then from 0 to now y equals 1 minus x...

Question

(1 point) Express the volume of the solid under the plane x - 3y + z = 6 and above the region bounded by 2x + y = 4 and x² + y = 4. Volume = \int_a^b \int_c^d \text{} \, dy \, dz where a = \text{} b = \text{} c = \text{} d = \text{} Find the volume. Volume = \text{}

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