How do you use a triple integral to find the volume of the given the tetrahedron enclosed by the coordinate planes 2x+y+z=3?
we can start by writing the plane as
so by setting z = 0, we know that the place cuts across the xy plane on 2x + y = 3
by setting (x,y) = 0, we know that the plane hits the z axis at z = 3
so we can draw it all in the first octant
The triple Integral is
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To find the volume of the tetrahedron enclosed by the coordinate planes and the plane (2x + y + z = 3), you can set up a triple integral in the appropriate coordinate system. Since the tetrahedron is bounded by the coordinate planes, the limits of integration are determined by the intersection points of the plane with the coordinate axes.

Identify the limits of integration for each variable. For (x), (y), and (z), these limits will be determined by the intersection points of the plane with the coordinate axes.

Set up the triple integral using the appropriate limits of integration and integrand.

Integrate the function over the specified limits to find the volume of the tetrahedron.
The triple integral setup for finding the volume ((V)) of the tetrahedron can be expressed as:
[ V = \iiint_{\text{Region}} dV ]
Where (dV) represents a small volume element within the tetrahedron, and the limits of integration are determined by the intersection points of the plane with the coordinate axes.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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