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**1. Find the volume, using triple integrals, of the region in the first octant beneath the plane 2x+3y+2z = 6**

**2. http://tutorial.math.lamar.edu/Classes/CalcIII/TripleIntegrals.aspx**

**SOLUTION:**

1. Assume X and Y are 0. Solve for Z: 2(0)+3(0)+2z=6 => z=3 (0,0,3)

2. Assume X and Z are 0. Solve for Y: 2(0)+3(y)+2(0)=6 => y=2 (0,2,0)

3. Assume Y and Z are 0. Solve for X: 2(x)+3(0)+2(0)=6 => x=3 (3,0,0)

4. The plane in question passes through three points: (0,0,3), (0,2,0), (3,0,0). Find two vectors parallel to this plane.

a. V = (3,0,0) - (0,2,0) = (3,-2,0)

b. W = (3,0,0) - (0,0,3) = (3,0,-3)

5. Compute a Normal vector N of this plane with the two vectors obtained in the previous two steps.

a. N = v*w = (3,-2,0) [(3,0,-3)]

b. i.=6, j.= - 9 and k.=6

c. (6x-9y+6z) = ((x-3)i+yj+zk) => 6x-9y+6z=18 => Solve for Z: z= -x+3y/2+3

d. Solve for y => (using the slope equation: (0-2)/(3-0)=> -2x/3+2 =y

e. Solve for X => We already know that it is 3.

f. Plug in x, y, and z information into the formula listed above.

e. My answer is 9, but the [final exam review for this problem is 3]

Please help. I am taking my final this week to finish my summer school before school starts next week.

Thank you all for your help.

DOH-WLcat.

1. Assume X and Y are 0. Solve for Z: 2(0)+3(0)+2z=6 => z=3 (0,0,3)

2. Assume X and Z are 0. Solve for Y: 2(0)+3(y)+2(0)=6 => y=2 (0,2,0)

3. Assume Y and Z are 0. Solve for X: 2(x)+3(0)+2(0)=6 => x=3 (3,0,0)

4. The plane in question passes through three points: (0,0,3), (0,2,0), (3,0,0). Find two vectors parallel to this plane.

a. V = (3,0,0) - (0,2,0) = (3,-2,0)

b. W = (3,0,0) - (0,0,3) = (3,0,-3)

5. Compute a Normal vector N of this plane with the two vectors obtained in the previous two steps.

a. N = v*w = (3,-2,0) [(3,0,-3)]

b. i.=6, j.= - 9 and k.=6

c. (6x-9y+6z) = ((x-3)i+yj+zk) => 6x-9y+6z=18 => Solve for Z: z= -x+3y/2+3

d. Solve for y => (using the slope equation: (0-2)/(3-0)=> -2x/3+2 =y

e. Solve for X => We already know that it is 3.

f. Plug in x, y, and z information into the formula listed above.

e. My answer is 9, but the [final exam review for this problem is 3]

Please help. I am taking my final this week to finish my summer school before school starts next week.

Thank you all for your help.

DOH-WLcat.