Multiple integrals: Find the volume bounded by the following surfaces

In summary, the task is to find the volume bounded by a plane, a paraboloid, and other planes. This can be accomplished through a triple integral, with the order of integration determined by the most complex limit. The range for z is 0 to 10x^2 + 4y^2, for y it is 14 to 2x, and for x it is determined by plotting the known facts involving x and y. The integrand for this volume is 1.
  • #1
ohlala191785
18
0

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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  • #2
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.
 
  • #3
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!
 
  • #4
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.
 
  • #5
OK I will try plotting. Thanks for the help.
 

Related to Multiple integrals: Find the volume bounded by the following surfaces

1. What is a multiple integral?

A multiple integral is an extension of a single integral to two or more dimensions. It is used to calculate the area, volume, or other higher-dimensional quantities of a region bounded by multiple surfaces.

2. How do you find the volume bounded by multiple surfaces using integrals?

To find the volume bounded by multiple surfaces, you can use the triple integral, which involves integrating a function over a three-dimensional region. This integral can be broken down into smaller integrals by using the properties of single and double integrals.

3. What are the limits of integration for multiple integrals?

The limits of integration for multiple integrals depend on the type of region being considered. For rectangular regions, the limits will be constant values for each variable. For more complex regions, the limits will be functions of one or more variables.

4. What is the difference between a double integral and a triple integral?

A double integral is used to calculate the area of a two-dimensional region, while a triple integral is used to calculate the volume of a three-dimensional region. In a triple integral, the limits of integration will be in terms of three variables, while in a double integral, the limits will be in terms of two variables.

5. Can multiple integrals be used for regions with curved surfaces?

Yes, multiple integrals can be used to calculate the volume of regions with curved surfaces. In this case, the limits of integration will be in terms of equations that define the curved surfaces, and the integrand will be a function of all variables involved.

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