# How do you find the volume of the solid bounded by the coordinate planes and the plane #8x + 6y + z = 6#?

use double integral (or triple if you like, i'll just do double as triple here is just extra unnecessary formality)

first we need to find the volume in question.

it hits the x,y,z axes as follows

so we can start with a drawing!!

so it's just a case now of finding the integration limits for this double integral

in the x-y plane we have

so either of these is fine

Either way, I get answer 3/4

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To find the volume of the solid bounded by the coordinate planes and the plane (8x + 6y + z = 6), we can use the method of triple integration with appropriate limits.

The volume (V) can be expressed as a triple integral over the region (D) in the (xyz)-space bounded by the planes (x = 0), (y = 0), (z = 0), and (8x + 6y + z = 6):

[ V = \iiint_D dV ]

To set up the limits of integration, we first find the intersection points of the plane with the coordinate planes:

When (x = 0), (y = 0), (z = 0), and (8x + 6y + z = 6), we get the points ((0, 0, 6)), ((0, 1, 0)), and ((\frac{3}{4}, 0, 0)), respectively.

Now, the limits of integration for (x), (y), and (z) are as follows:

For (x): (0 \leq x \leq \frac{3}{4})

For (y): (0 \leq y \leq 1 - \frac{4}{3}x)

For (z): (0 \leq z \leq 6 - 8x - 6y)

So, the volume (V) can be computed by evaluating the triple integral:

[ V = \int_0^{\frac{3}{4}} \int_0^{1 - \frac{4}{3}x} \int_0^{6 - 8x - 6y} dz , dy , dx ]

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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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