Step by step
The question
Question: find the volume of the solid under the plane x + y + 2z = 4 in the first octant.
1 Draw it
First octant means x, y and z are all positive: the corner in front.
The plane cuts the axes at x = 4, y = 4 and z = 2.
The solid is the corner piece under it.
Its floor is the green triangle.
2 Find the limits
z goes from the floor, z = 0, up to the plane: solve the plane for z.
On the floor, z = 0, so the edge is x + y = 4.
Solve for y: y goes from 0 to 4 − x.
And x goes from 0 to 4, where that edge meets the x-axis.
Put the limits in: the most changing ones inside.
Careful!
The inside limits may have letters.
The outside limits must be plain numbers.
3 The inner integral
Inside first: integrate in z.
x and y stay fixed, like plain numbers.
Integrate each term: the power goes up by 1, and we divide by the new power.
Put in the top z = 2 − x/2 − y/2, then take away the bottom z = 0.
Work out each part.
The bottom part is 0.
So it is 2 − x/2 − y/2.
4 The middle integral
Then the middle one: integrate in y.
x stays fixed.
Integrate each term: the power goes up by 1, and we divide by the new power.
Put in the top y = 4 − x, then take away the bottom y = 0.
Work out each term on its own: multiply out the brackets.
Add the like terms in each part.
The bottom part is 0.
So it is 4 − 2x + x² over 4.
5 The outer integral
Last the outside one, in x.
Its limits are numbers.
Integrate each term: the power goes up by 1, and we divide by the new power.
Put in the top x = 4, then take away the bottom x = 0.
The bottom part is 0.
So it is 16/3.
6 Check
Let's check.
The solid is a pyramid: one third of the base times the height.
The base triangle is half of 4 times 4.
We get 16/3.
Right!
Let's look back.
First: draw it.
Then: find the limits.
Then: the inner integral.
Then: the middle integral.
Then: the outer integral.
Last: check.