Write six different iterated triple integrals for the volume...

Write six different iterated triple integrals for the volume of the tetrahedron cut from the first octant by the plane 6x + 3y + 2z = 6. Evaluate the first integral. Write an integral that represents the volume in the order dx dy dz. $$int_0^square int_0^square int_0^square square dx dy dz$$

Transcript

00:01 So here it is given that the first obtained by the plane of 6x plus 3y plus 2z is equals to 6 we have to evolve with the first integral.

00:14 So we can observe this volume starting with every one of the three variables and the second and the other can or be any of remaining two variables.

00:27 So therefore there are total three into two, that is the sixth different ways to write the wanted volume.

00:38 So in order to start with determining the bounds for the first integral of the three, we must find the intersection of given plane and all three axes.

00:54 So we find the intersection with, for example, let's say the x -axis, is plugging in the values of y equals to zero and z is equal to zero in the given equation.

01:10 So for the x intersection, we plug in the y equals to zero and z equals to zero, we get a six x plus three times of zero plus the two times of zero will come out equals to six that is equal to six.

01:26 This implies that the x is equals to one.

01:30 Similarly for the y intersection we put x as 0 and z as 0 so this will become 3y plus 2 times of 0 will be equal to how much that is 3y is equal to 6 so from here y will come at equal to 2 similarly for z intersection we put x as 0 that is 6 into 0 plus 3 into y 0 plus the 2 times of z will be equal to 6 comes out z is equal to 3 so there's are the intercepts x y and z now after we place the outer integral within have to choose the two of the middle one so to know the bounds of that middle integrals we need to determine that restriction between the given flame and each of the planes that is o of y, okay, x of y and x of zate and the third plane you will say x of z.

02:37 So let's see, we want to write the first presentation like integration into the bracket integration into the vagrant integration of 1dx, d, y, and z.

02:48 So from here for z of y plane, we can clearly say that let's say x z equals to zero...

Question

Write six different iterated triple integrals for the volume of the tetrahedron cut from the first octant by the plane 6x + 3y + 2z = 6. Evaluate the first integral. Write an integral that represents the volume in the order dx dy dz. $$int_0^square int_0^square int_0^square square dx dy dz$$

Please give Ace some feedback