# Determining Volume using double integral

• cse63146
In summary, to find the volume of the solid in the first octant bounded by the planes z = x + y + 1 and z = 5 - x - y, you would set up a triple integral by drawing the region in the z=0 plane and integrating z*dx*dy over that region. The integral of 1*dz is then taken over the volume. To accurately visualize the region, it is important to first determine where the two planes intersect.
cse63146

## Homework Statement

Find the volume of the solid in the first octant by the planes z = x + y + 1 and z = 5 - x - y

## The Attempt at a Solution

How would I set this up?

You would set it up by drawing the region in the z=0 (i.e. xy plane) and integrating z*dx*dy over that region.

Are you sure it's not supposed to be dz dy dx? I went to my prof with this one, and the way he set it up as a series of triple integrals.

Your post title was "Determining Volume using double integral". It you want to set it up as a triple integral first then you integrate 1*dz*dx*dy over the volume. The integral of 1*dz is just z. It's the same thing. Now get started.

http://img16.imageshack.us/img16/1340/39161133fw1.jpg

Its not to scale though; I just need to worry about the top right quadrant.

$$\int^{5}_{0}\int^{x - 5}_{0}\int^{5- x - y}_{0} dz dy dx + \int^{1}_{0}\int^{-x - 1}_{0}\int^{x + y +1}_{0} dz dy dx$$

How off am I?

Last edited by a moderator:
Not too good I don't think. I'm not too sure what you are trying to draw, but your first step should be to figure out where the two planes intersect. Can you do that? I think that will help you picture the region.

## 1. What is the concept behind using double integrals to determine volume?

The concept behind using double integrals to determine volume is to divide the 3D object into infinitesimal slices and integrate over each slice to find the total volume. This is similar to how regular integrals are used to find the area under a curve in 2D.

## 2. How do you set up a double integral for determining volume?

To set up a double integral for determining volume, you need to first identify the bounds of integration for both the x and y variables. These bounds are determined by the shape and size of the 3D object. Then, you need to determine the function that represents the height of the object at each point in the x-y plane. This function will be the integrand in the double integral.

## 3. Can double integrals be used for any shape or size of 3D object?

Yes, double integrals can be used to determine the volume of any shape or size of 3D object. As long as the object can be divided into infinitesimal slices and the height function can be determined for each slice, double integrals can be used.

## 4. What are some real-world applications of determining volume using double integrals?

Determining volume using double integrals has many real-world applications. It is commonly used in physics and engineering to calculate the volume of irregularly shaped objects such as fluids or structural components. It is also used in economics and finance to calculate the volume of complex financial instruments.

## 5. Are there any alternative methods for determining volume besides using double integrals?

Yes, there are alternative methods for determining volume such as using triple integrals, which can be used for more complex 3D objects. Another method is to use geometric formulas for specific shapes, such as the formula for the volume of a sphere or cylinder. However, double integrals are a versatile and efficient method for determining volume in many cases.

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