Double Integral set up problem

In summary, the problem is to set up an iterated integral for the volume of the region between the plane z=5 and the graph of f(x,y)=21-(x^2+y^2)^2. The bounds of integration should be the level sets of the function, with the lower bound at the c=5 level set and the upper bound at the c=0 level set. The function to be integrated should be f(x,y) - 5 over the region where f(x,y) > 5.
  • #1
efekwulsemmay
54
0

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?
 
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  • #2
hi efekwulsemmay! :smile:
efekwulsemmay said:
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 :smile:​
 
  • #3
tiny-tim said:
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)
 
  • #4
efekwulsemmay said:

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 [itex]f(x,y) - 5[/itex] over the region where [itex]f(x,y) > 5[/itex].
 
  • #5
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?
 

FAQ: Double Integral set up problem

1. What is a double integral set up problem?

A double integral set up problem is a mathematical problem in which you are given a two-dimensional region and are asked to calculate the volume under a three-dimensional surface within that region. It requires the use of two integrals, one for each variable, and involves understanding the limits of integration and the function being integrated.

2. How do I set up a double integral problem?

To set up a double integral problem, you first need to identify the limits of integration for both variables, which are typically given by the boundaries of the region. Then, you need to determine the function being integrated and write it in terms of both variables. Finally, you can set up the double integral by placing the function between the two limits of integration and integrating with respect to one variable first, and then the other.

3. What is the purpose of solving a double integral set up problem?

The purpose of solving a double integral set up problem is to calculate the volume under a three-dimensional surface within a given two-dimensional region. This is useful in many scientific and engineering fields, such as physics, engineering, and economics, as it allows for the determination of important quantities such as mass, density, and probability.

4. What are the challenges of solving a double integral set up problem?

The main challenges of solving a double integral set up problem include correctly identifying the limits of integration and understanding the function being integrated. Additionally, setting up the integral correctly and performing the integration can also be challenging for some people. Practice and familiarity with the concept can help overcome these challenges.

5. What are some common applications of double integral set up problems in science?

Double integral set up problems have many applications in science, including calculating the volume of irregularly shaped objects, finding the center of mass of a two-dimensional region, determining the probability of an event in statistics, and calculating the electric field in physics. They are also used in many real-world situations, such as calculating the area of a field for agricultural purposes or determining the optimal production level for a company in economics.

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