Multiple integrals: Find the volume bounded by the following surfaces

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Homework Help Overview

The problem involves finding the volume bounded by specific planes and a paraboloid, with the surfaces defined by equations such as z = 0, x = 0, y = 2x, y = 14, and z = 10x^2 + 4y^2.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the use of triple integrals and the importance of determining the correct order of integration. Questions arise regarding the limits for x, particularly how to establish the upper bound, and whether the integrand should simply be 1.

Discussion Status

Some participants have provided guidance on setting up the triple integral and suggested plotting to visualize the bounds. There is an ongoing exploration of the limits for the variables involved, particularly x.

Contextual Notes

Participants are working under the constraints of homework rules, which may limit the amount of direct assistance provided. The discussion reflects uncertainty about the limits of integration and the setup of the problem.

ohlala191785
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Homework Statement



Find the volume bounded by the following surfaces:
z = 0 (plane)
x = 0 (plane)
y = 2x (plane)
y = 14 (plane)
z = 10x^2 + 4y^2 (paraboloid)

Homework Equations



The above.

The Attempt at a Solution



I think it has something to do with triple integrals? But I have no idea how to approach this (e.g. what the limits are, what to integrate, etc.)

Any help would be greatly appreciated!
 
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Yes, triple integral is fine. The trick is to get them in the best order. Generally start with the one that has the most complicated limit, in this case z.
The first integral can have a range that depends on all the other variables, the second on all except that one, and so on.
So try to write out:
- the range for z, given x and y
- the full range for y, given x only
- the full range for x, regardless of y, z.
 
So z is from 0 to 10x^2 + 4y^2, y from 14 to 2x, but what would the limit of x be? The lower bound is 0, how would I find the higher one? Also is the integrand just 1?

Thanks!
 
ohlala191785 said:
So z is from 0 to 10x^2 + 4y^2, y from 14 to 2x, but what would the limit of x be? The lower bound is 0, how would I find the higher one?
Plot the known facts involving only x and y. You'll soon see what the range for x is.
Also is the integrand just 1?
Yes.
 
OK I will try plotting. Thanks for the help.
 

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