Iterated integral questions

[Bob19]

I need to find the volume of solid bounded by the elliptic paraboloid z = 1 + (x-1)^2 + 4y^2, the planes x = 3 and y = 2 and the coordinate planes.


I claim that the solid called S is bounded above the square R = [0,3] x [0,2]

Thereby resulting in a volume V = 44.

Is that a correct assumption ?

Sincerely and best regards,

Bob

 

[Fermat]

I check your answers.

 

[tacman]

Hello Bob,

Thank you for your question. To answer your question, yes, your assumption is correct. The solid S is indeed bounded by the given elliptic paraboloid and the coordinate planes, as well as the planes x = 3 and y = 2.

To find the volume of this solid, we can use the formula for a double integral over a rectangular region R:

V = ∫∫S dV = ∫∫R f(x,y) dA

In this case, the function f(x,y) is given by the equation for the elliptic paraboloid z = 1 + (x-1)^2 + 4y^2. So, we have:

V = ∫∫R (1 + (x-1)^2 + 4y^2) dA

Substituting in the bounds for the region R, we have:

V = ∫0^3 ∫0^2 (1 + (x-1)^2 + 4y^2) dy dx

Evaluating this integral gives us a volume of 44, which confirms your assumption.

I hope this helps clarify the solution for finding the volume of the given solid. Let me know if you have any further questions or concerns.

Best regards,