# How to find the limits of a volume integral?

• Hawkingo
In summary: The volume integral you have left is just the integral of the volume element. This gives the volume enclosed by the surface, ie, the volume of a sphere in this case.
Hawkingo

## Homework Statement

If ##\vec { F } = x \hat { i } + y \hat { j } + z \hat { k }## then find the value of ##\int \int _ { S } \vec { F } \cdot \hat { n } d s## where S is the sphere ##x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4##.

## The Attempt at a Solution

From gauss divergence theorem we know
##\int \int _ { S } \vec { F } \cdot \hat { n } d s = \int \int \int _ { V } \vec { \nabla } \cdot \vec { F } d v##
So ##\int \int _ { S } \vec { F } \cdot \hat { n } d s = \int \int \int _ { V } [ \frac { \partial } { \partial x } ( x ) + \frac { \partial } { \partial y } ( y ) + \frac { \partial } { \partial z } ( z ) ] d x d y d z##
=##3 \int \int \int _ { V } d x d y d z##

Here if I'll calculate the answer using gauss divergence theorem, then I have to find the volume integral of the function, but how will I take the limits of the integration. Can someone help please?
Thank you.

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The volume integral you have left is just the integral of the volume element. This gives the volume enclosed by the surface, ie, the volume of a sphere in this case.

Hawkingo
Orodruin said:
The volume integral you have left is just the integral of the volume element. This gives the volume enclosed by the surface, ie, the volume of a sphere in this case.
Thanks, I was confused because the book solved in a very long and difficult manner and the limits used were seemed to be inappropriate, but by your way I arrived at the answer in just one step. Thanks a lot

Orodruin said:
The volume integral you have left is just the integral of the volume element. This gives the volume enclosed by the surface, ie, the volume of a sphere in this case.
It was solved because the equation was of a sphere, but if it was something unknown can you suggest me a way how to derive the limits from the equation without knowing the figure?

Hawkingo said:
It was solved because the equation was of a sphere, but if it was something unknown can you suggest me a way how to derive the limits from the equation without knowing the figure?
Using the figure is the way to determine the limits because the coordinates are intimately related to the geometry.

Hawkingo said:
It was solved because the equation was of a sphere, but if it was something unknown can you suggest me a way how to derive the limits from the equation without knowing the figure?
Presumably you've already covered this in your class. There is no cookbook way of coming up with the limits. Each problem is different.

You should be able to do this problem by evaluating the three-dimensional integral. Maybe you should start there, since you know what the answer is supposed to be.

Hawkingo said:

## Homework Statement

If ##\vec { F } = x \hat { i } + y \hat { j } + z \hat { k }## then find the value of ##\int \int _ { S } \vec { F } \cdot \hat { n } d s## where S is the sphere ##x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4##.

## The Attempt at a Solution

From gauss divergence theorem we know
##\int \int _ { S } \vec { F } \cdot \hat { n } d s = \int \int \int _ { V } \vec { \nabla } \cdot \vec { F } d v##
So ##\int \int _ { S } \vec { F } \cdot \hat { n } d s = \int \int \int _ { V } [ \frac { \partial } { \partial x } ( x ) + \frac { \partial } { \partial y } ( y ) + \frac { \partial } { \partial z } ( z ) ] d x d y d z##
=##3 \int \int \int _ { V } d x d y d z##

Here if I'll calculate the answer using gauss divergence theorem, then I have to find the volume integral of the function, but how will I take the limits of the integration. Can someone help please?
Thank you.

Another way is to note that ##\vec{F} \cdot \vec{n} = r## = radius of the sphere, so your integral is ##\int \!\! \int_S r \, dS = r \int \! \! \int_S dS.## Do you recognize that last integral?

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## 1. How do you determine the limits of a volume integral?

The limits of a volume integral can be determined by considering the boundaries of the region being integrated. This can be done by examining the given function or by graphing the region and identifying the points of intersection.

## 2. What is the importance of finding the limits of a volume integral?

Finding the limits of a volume integral is important because it allows us to accurately calculate the volume of a three-dimensional region. It also helps us to understand the boundaries of the region and how the function changes within those boundaries.

## 3. Can the limits of a volume integral change?

Yes, the limits of a volume integral can change depending on the region being integrated. They can also vary depending on the coordinate system being used.

## 4. How can I check if my limits of integration are correct?

You can check if your limits of integration are correct by evaluating the integral using the given limits and comparing it to the known volume of the region. If they match, then the limits are correct.

## 5. Are there any shortcuts for finding the limits of a volume integral?

There are no shortcuts for finding the limits of a volume integral. It requires careful consideration and understanding of the region being integrated and the given function.

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