# Calculating the Volume of a Tetrahedron using Integration Method

• paraboloid
In summary, the conversation discusses the process of finding the volume of a tetrahedron using integration. The correct setup of the integral is given, but there is a mistake in the arithmetic calculation which needs to be corrected. The importance of learning Tex for easier reading is also mentioned.

#### paraboloid

Volume of Tetrahedron[Solved]

My textbook opts to integrate with respect to y before x(dydx vs dxdy), so I assumed that it would not affect the outcome.

I set the upper and lower bounds of y, respectively, as y = 24 - 7x/4 (from 7x+4y=96) to y = x/4 (from x = 4y). For x I set it from upper bound x = 12 (using x/4 = 24 - 7x/4) to lower bound x = 0 (given).

Integrating with respect to y, I get $$\int(96y - 7xy - 2y^2)|dx$$ which after inputing the bounds become $$\int(1152-144x+8x^2)dx$$. After integrating this I get $$(1152x - 72x^2 + 8x^3/3)|$$.
Computing this I get 1152(12) - 72(144) + 8(1728)/3 = 13824-10368+4608 = 8064.

I've been at this for a few hours now, but I can't seem to find my error.
Any help would be great,

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It would be good to learn a little tex to make it easier to read. Just click on the equation to see it:

$$\int_0^{12}\int_{\frac x 4}^{24 - \frac {7x}{4} }dydx$$

If I read your post correctly, this is how you set up the integral and it is correct. You just need to chase down the arithmetic error in your calculation, which, unfortunately, I don't have time right now to help you with. Good luck.

I get 8x^2 - 192x + 1152 after integrating w/ respect to y.

Thank you both! I'll definitely work on my latex once things settle down so that I don't cause so much confusion. And yes, in fact I add 24 to -168 instead of subtracting.

## What is the formula for calculating the volume of a tetrahedron?

The formula for calculating the volume of a tetrahedron is V = (1/3) * (base area) * (height), where the base area is equal to one-half the product of the base length and the base width, and the height is the perpendicular distance from the base to the opposite vertex.

## What is a tetrahedron?

A tetrahedron is a polyhedron with four triangular faces, six straight edges, and four vertices. It is a three-dimensional shape that resembles a pyramid with a triangular base.

## How do you find the height of a tetrahedron?

The height of a tetrahedron can be found by using the formula h = (√2/3) * (s), where s is the length of one of the edges. Alternatively, the height can also be found by dividing the volume of the tetrahedron by the area of the base.

## What is the relationship between the volume and surface area of a tetrahedron?

The volume of a tetrahedron is directly proportional to its surface area. This means that as the volume increases, the surface area also increases, and vice versa. However, the rate at which they change is not constant and depends on the shape and size of the tetrahedron.

## How is the volume of a tetrahedron different from other 3D shapes?

The volume of a tetrahedron is calculated differently from other 3D shapes because it has a unique shape and only four faces. Unlike cubes, cylinders, or spheres, which have well-defined formulas for finding their volumes, the volume of a tetrahedron involves finding the area of its base and multiplying it by the height.