Multiple integrals: Find the volume bounded by the following surfaces

[ohlala191785]

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!

 

[haruspex]

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.

 

[ohlala191785]

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!

 

[haruspex]

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.

 

[ohlala191785]

OK I will try plotting. Thanks for the help.