Volume under the elliptic paraboloid

In summary, the formula for calculating the volume under an elliptic paraboloid is V = (πabc)/3, and it is similar to a regular paraboloid but with an elliptical base. The volume cannot be negative, and it is commonly used in engineering and architecture. It can also be approximated using Cavalieri's principle.
  • #1
Damascus Road
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0
Greetings again,

this is another I'm stuck on...

Calculate the volume under the elliptic paraboloid z = x^{2} + 4y^{2} and over the rectangle R = [-1, 1] \times [-4, 4].

I'm not sure how to find the limits of z for the triple integral. Can I somehow integrate the function z, and only do a double integral? Perhaps going from -1, 0 and -4, 0 and then double it?
 
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  • #2
I believe I figured it out now, never mind.
 

1. What is the formula for calculating the volume under an elliptic paraboloid?

The formula for calculating the volume under an elliptic paraboloid is V = (πabc)/3, where a, b, and c are the semi-axes of the elliptical base.

2. How is the volume under an elliptic paraboloid different from a regular paraboloid?

The volume under an elliptic paraboloid is similar to a regular paraboloid, but the base of an elliptic paraboloid is an ellipse instead of a circle. This results in a more elongated shape and a larger volume compared to a regular paraboloid with the same height and base radius.

3. Can the volume under an elliptic paraboloid be negative?

No, the volume under an elliptic paraboloid cannot be negative. The volume represents the amount of space enclosed by the surface, and it is always a positive value.

4. How is the volume under an elliptic paraboloid used in real-life applications?

The volume under an elliptic paraboloid is commonly used in engineering and architecture for calculating the capacity of storage tanks, calculating the volume of a curved roof, and designing underground structures such as tunnels and reservoirs.

5. Is there a way to approximate the volume under an elliptic paraboloid without using the exact formula?

Yes, the volume under an elliptic paraboloid can be approximated by dividing the surface into smaller sections and using the formula for the volume of a frustum of a cone. This method is known as the Cavalieri's principle and can provide a good estimation of the volume under an elliptic paraboloid if the sections are small enough.

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