# How do you use the triple integral to find the volume of the solid in the first octant bounded by the coordinate planes and the plane 3x+6y+4z=12?

Graph the solid, determine the limits of integration, then integrate.

It is always useful (and usually necessary) to graph the solid you are trying to find the volume of in order to determine the limits of integration. This can be done quite easily by hand in this case.

You can graph the plane by finding the intercept for each axis and then simply connecting those points.

Setting

The given solid is thus bounded below by the

Here is a 3-dimensional graph of the given plane:

Here is a graph of the complete solid:

In this case, you can choose to integrate with respect to any order of the variables. I have chosen to follow the order

If we were to enter the solid along the

Thus, our upper limit of integration with respect to

We have now finished with

Now with respect to

Thus, our upper limit with respect to

Finally, we can easily determine the upper and lower bounds for

Thus, our upper limit with respect to

To find the volume of the solid, we keep the integrand at a value of

Evaluating for our limits of integration from

We now integrate with respect to y:

Evaluating for our limits of integration from

Thus, the volume of the solid is 4

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To find the volume of the solid in the first octant bounded by the coordinate planes and the plane (3x + 6y + 4z = 12), we can set up a triple integral over the region.

First, we need to determine the limits of integration for (x), (y), and (z) that define the region in the first octant bounded by the coordinate planes and the plane.

The plane intersects the coordinate axes at (x = 4), (y = 2), and (z = 3). So, the limits of integration are (0 \leq x \leq 4), (0 \leq y \leq 2), and (0 \leq z \leq 3).

The volume integral is then:

[ \iiint_{\text{Region}} dV ]

where (dV = dx , dy , dz).

Setting up the integral with the limits of integration:

[ \int_0^3 \int_0^2 \int_0^4 dx , dy , dz ]

Evaluating the integral:

[ \int_0^3 \int_0^2 4 , dy , dz ] [ \int_0^3 8 , dz ] [ 8 \cdot 3 ] [ 24 ]

So, the volume of the solid in the first octant bounded by the coordinate planes and the plane (3x + 6y + 4z = 12) is 24 cubic units.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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