Drawing a Region of Integration

In summary, the student is trying to solve a problem in calculus involving boundaries and is having difficulty because the variables x and y are not allowed in the outermost integral's bounds. If -3≤x≤0, the student is not in the first quadrant and c is the correct answer. If x=-3, the student can check the bounds to see if they are correct.
  • #1
TranscendArcu
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0

Homework Statement



http://img803.imageshack.us/img803/6397/skjermbilde20111130kl50.png

The Attempt at a Solution

I seriously can't even figure out how this region is drawn. I think in the second integral I need y to range from -x to 3. Then the answer could conceivably be a). But, as is, I can't do this.
 
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  • #2
Hi TranscendArcu! :smile:Perhaps you can start with a drawing in which you mark the end points of the boundaries?
And if you have that, a couple of points in between?

If you switch the integrals around, the outer integral(s) will be with respect to y.
y can not be dependent on x then, since x does not exist anymore when you integrate y.
The outer integral should typically cover the largest possible range, while the inner integral sets a condition on what to integrate.
 
  • #3
Okay. Here's what I have: in the first quadrant, a triangle with vertices (0,0), (3,3), (0,3). This is the region I get when I map out the region described by the first integral. In the second integral I have x≤y≤3, which is exactly the same constraint on y I had before. If I let -3≤x≤0, in the first quadrant I have a square with vertices (0,0), (0,3), (-3,3), (-3,0). In the third quadrant I have another triangle with vertices (-3,-3), (0,0), (-3,0).

!

Is it c?
 
  • #4
Hmm, before saying yeah or nay, let's explore your analysis.

If -3≤x≤0, then you're not in the first quadrant...
Furthermore, if x=-3, can you recheck the bounds?So you think it's c?
What is your analysis then of the region that c covers?
 
  • #5
I think I mistyped something: If I let -3≤x≤0, in the second quadrant I have a square with vertices (0,0), (0,3), (-3,3), (-3,0).

But I still think it's c, since my picture did actually have the square in Q2.
 
  • #6
All right!

So your problem gives you 2 triangles and a square.
And c gives you a square in Q2.

Are these the same figures?
 
  • #7
I think so. If I graph the bounds in c I seem to get the same region.
 
  • #8
And what about a, b, and d?
 
  • #9
a is a triangle in the first and second quadrants. b I found more tricky, but I think it's the region in (a) and also its reflection over the x-axis. Who knows what d is? As you said in #2, it doesn't make sense to have variables in the outermost integral's bounds.
 
  • #10
Sounds like you've got things down!

So?
 
  • #11
You still seem unsure...?
I was kind of hoping for a statement like: "I'm sure it is c!".
 
  • #12
Pick c! I was pretty sure it was c after I really mapped out the region being described. Just by inspection, c seemed to be the appropriate answer once graphed correctly. Still, it took me a minute or two to realize what region b described.
 
  • #13
Good! :smile:
 

FAQ: Drawing a Region of Integration

What is a region of integration?

A region of integration is a two-dimensional space that is enclosed by a set of boundaries, such as curves or lines. It is used in mathematics and physics to calculate the area under a curve or the volume of a three-dimensional object.

How do you determine the limits of integration for a given region?

The limits of integration are determined by the boundaries of the region. These boundaries can be expressed as equations in terms of x and y, and the limits are then found by solving these equations simultaneously. Alternatively, the limits can also be determined graphically by plotting the region and finding the points of intersection.

What is the difference between a single and double integral?

A single integral is used to calculate the area under a curve in one dimension, while a double integral is used to calculate the volume under a surface in two dimensions. A double integral involves two sets of limits, one for each dimension, while a single integral only has one set of limits.

How does the choice of coordinate system affect the region of integration?

The choice of coordinate system can affect the region of integration as it determines how the boundaries of the region are expressed. For example, in polar coordinates, the boundaries may be expressed in terms of angles and radii, while in Cartesian coordinates, the boundaries are expressed as equations in terms of x and y.

What are some common methods for drawing a region of integration?

Some common methods for drawing a region of integration include graphing the region, using symmetry properties to simplify the region, and using transformations to change the region into a simpler form. Additionally, technology such as graphing calculators or computer software can also be used to visualize and draw the region of integration.

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