Double Integral set up problem

[efekwulsemmay]

Homework Statement

(exact wording from my homework set) Set up an iterated integral for the volume of the region which is above the plane z=5 and below the graph of f(x,y)=21-(x^2+y^2)^2. Pay attention to what the region of integration should be!

Homework Equations

Not sure.

The Attempt at a Solution

Ok so I figured out the equation for the lower bound of the volume. It's going to be the circle

x^2+y^2=4

I can visualize the shape of the object I just can't figure out where this lower bound is supposed to be in the limits of integration and I am not sure how to describe the upper limit of the object. I know it's the graph but how does that fit into the limits? Another question is what equation am I going to be integrating? Should it be the function given in the problem?

 

[tiny-tim]

hi efekwulsemmay!

Set up an iterated integral for the volume of the region which is above the plane z=5 and below the graph of f(x,y)=21-(x^2+y^2)^2.
…
I can visualize the shape of the object I just can't figure out where this lower bound is supposed to be in the limits of integration and I am not sure how to describe the upper limit of the object. I know it's the graph but how does that fit into the limits? Another question is what equation am I going to be integrating? Should it be the function given in the problem?

you want the volume, so you are integrating 1 times dxdydz over the whole volume

just decide which variable you're going to integrate first (x, y, or z?), and then decide what the limits are

then do the second variable, then do the third variable …

show us how far you get ​

 

[efekwulsemmay]

you want the volume, so you are integrating 1 times dxdydz over the whole volume

But I thought that since it is a two variable function that it would have to be a double integral. (Also we haven't covered triple integrals in class yet so I don't think he'd assign a problem like that on the homework)

 

[pasmith]

Homework Statement

(exact wording from my homework set) Set up an iterated integral for the volume of the region which is above the plane z=5 and below the graph of f(x,y)=21-(x^2+y^2)^2. Pay attention to what the region of integration should be!

Homework Equations

Not sure.

The Attempt at a Solution

Ok so I figured out the equation for the lower bound of the volume. It's going to be the circle

x^2+y^2=4

I can visualize the shape of the object I just can't figure out where this lower bound is supposed to be in the limits of integration and I am not sure how to describe the upper limit of the object. I know it's the graph but how does that fit into the limits? Another question is what equation am I going to be integrating? Should it be the function given in the problem?

The volume between two surfaces is the integral of the upper surface less the integral of the lower surface. So you need to integrate $f(x,y) - 5$ over the region where $f(x,y) > 5$.

 

[efekwulsemmay]

Would it make sense to think about the bounds of integration as the level sets of the function? As in, I need to integrate from the c=5 level set to the c=0 level set (since the c=0 level set gives the maximum value for the function)?

And then to use that as the bounds for the double integral?