Surface integral or Divergence Theorem confused?

In summary, the task is to find the volume of the region bounded by the line y=x-1 and the parabola y^2=2x+6 using the integral ∫∫ xy DA. The correct method is to find the intersection of the two equations, use those points as limits for the integral, and solve to get a final answer of 52/3. However, there may be some confusion about the use of the word "volume" in this context, as the region is two-dimensional and should technically have an area rather than a volume. The correct integral to use in this case would be ∫∫ |xy| dA.
  • #1
abrowaqas
114
0

Homework Statement



Find the Volume
∫∫ xy DA
where R is the region bounded by by the line y=x-1 and the parabola y^2=2x+6.

Homework Equations



∫∫ xy dx dy

The Attempt at a Solution



first i found the intersection of the above equations . which is (5,4) to (-1,-2) . then i simple put the values in the limits of the integral
i-e

∫(from y= -2 to y=4) ∫(from x= y+1 to x= (y^2-6)/2 ) xy dx dy

and solve it and finaly got 52/3.

is this the right method and limits are correct or not ?
or i have to use here divergence theorem

can somebody explain the word VOLUME in the question ?
 
Physics news on Phys.org
  • #2
The word "volume" is an error. Clearly that is a two dimensional figure so it has area not volume.
 
  • #3
so that means my method is correct?

but someone says

If the problem really is to find a "volume", it should read ∫ ∫ |xy| dA because the integrand is not positive everywhere on the region given.

Can't be sure, but that's probably not what the author intended. He just wants you to integrate xy over the region. It has the form ∫ ∫ z(x, y) dA which is a volume integral if z(x, y) is non-negative over the region.
 

1. What is the difference between a surface integral and the Divergence Theorem?

A surface integral is a mathematical calculation that involves integrating a function over a surface, while the Divergence Theorem is a mathematical theorem that relates the integral of a vector field over a closed surface to the volume integral of the divergence of that vector field over the volume enclosed by the surface. In simpler terms, the Divergence Theorem allows us to calculate a surface integral using a volume integral.

2. How do I know when to use a surface integral or the Divergence Theorem?

You will typically use a surface integral when you need to calculate the flux of a vector field through a given surface, while the Divergence Theorem is useful when you need to relate the behavior of a vector field inside a closed surface to its behavior on the boundary of that surface.

3. Are there any prerequisites for understanding surface integrals and the Divergence Theorem?

Yes, a basic understanding of vector calculus and multivariable calculus is necessary in order to understand and apply surface integrals and the Divergence Theorem. Additionally, knowledge of the properties of vector fields, such as divergence and curl, is helpful.

4. Can the Divergence Theorem be applied to any vector field?

Yes, the Divergence Theorem can be applied to any continuous vector field in three-dimensional space. However, in some cases, it may be necessary to modify the theorem to account for certain conditions, such as a non-conservative vector field.

5. How can I visualize and understand the concept of a surface integral?

A surface integral can be visualized as the sum of infinitely small areas on a surface, where the value of the function being integrated is multiplied by the area of each small section. This calculation gives us the total amount of a quantity, such as flux, passing through the surface. Additionally, using three-dimensional graphs and visualizing the behavior of the vector field can aid in understanding the concept of a surface integral.

Similar threads

  • Calculus and Beyond Homework Help
Replies
10
Views
289
  • Calculus and Beyond Homework Help
Replies
3
Views
119
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
449
  • Calculus and Beyond Homework Help
Replies
6
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
988
  • Calculus and Beyond Homework Help
Replies
2
Views
480
  • Calculus and Beyond Homework Help
Replies
12
Views
936
  • Calculus and Beyond Homework Help
Replies
20
Views
385
  • Calculus and Beyond Homework Help
Replies
4
Views
809
Back
Top