# Volume integral, am I doing this right?

In summary, the conversation discusses how to evaluate the triple integral of a vector function V over a specific region using the divergence theorem. The dot product of V with the normal of the first surface is found to be zero, and the limits for the double integral over the second surface are determined to be from x=0 to x=√(6-y^2) and from y=0 to y=√6, as the surface is a circle with radius √6 in the xy-plane.
Hi all,

evaluating $$\int\int\int \nabla . V d\tau$$ over

$$x^{2} + y^{2} \leq 6, 0 \leq z \leq 10$$

where V is a vector function of just $$\hat{x}$$ and $$\hat{y}$$.

Using the divergence theorem, and doing the dot product of V with the normal of the first surface,

the two partials w.r.t x and y are 2x and 2y,
doing the cross product, the normal is 4xy z.

So, the dot product is zero since there is no z component to V?

Then, when I go to do the same for the second surface, I'm not sure what my double integral limits are, since I only have a surface of z...

help!
thanks!

Hello,

Thank you for your post. I understand your confusion regarding the limits of the double integral for the second surface. Let me try to clarify it for you.

Firstly, you are correct in your understanding that the dot product of V with the normal of the first surface is zero since there is no z component to V. This is because the vector function V only has components in the x and y directions.

Now, for the second surface, the limits of the double integral will depend on the specific shape of the surface. However, since the surface is defined by z=0, the limits of the double integral will be from x=0 to x=√(6-y^2) and from y=0 to y=√6. This is because the surface is a circle in the xy-plane with radius √6. Therefore, the double integral will be evaluated over this circular surface.

I hope this helps clarify things for you. If you have any further questions, please feel free to ask. Thank you.

## 1. How do I calculate the volume integral?

To calculate the volume integral, you need to integrate the function over the three-dimensional region. This can be done using a variety of methods, such as the triple integral or using divergence theorem.

## 2. What is the purpose of a volume integral?

A volume integral is used to calculate the volume of a three-dimensional object or region. It is commonly used in physics and engineering to determine properties such as mass, density, and electric charge.

## 3. How do I know if I am setting up the volume integral correctly?

To ensure that you are setting up the volume integral correctly, you should first identify the boundaries of the three-dimensional region and determine the correct order of integration. It is also helpful to graph the region and visualize the integration process.

## 4. What are some common mistakes when solving a volume integral?

Some common mistakes when solving a volume integral include incorrect identification of boundaries, incorrect order of integration, and mistakes in the integration process. It is important to carefully check your work and double-check your calculations to avoid these mistakes.

## 5. Are there any tips for simplifying a volume integral?

One tip for simplifying a volume integral is to identify symmetries in the region or function being integrated, as this can help reduce the number of integrals needed. Additionally, using appropriate substitutions and techniques such as cylindrical or spherical coordinates can also simplify the integral.

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