Converting to a Double Integral

Oct 13, 2025 by talynt.org 113 views

Introduction

In mathematics, finding the volume of a solid object is a fundamental problem that has numerous applications in various fields, including physics, engineering, and architecture. One of the most common methods for finding the volume of a solid is by using the concept of integration. In this article, we will discuss how to find the volume of a column with a square base cut by a plane using the method of integration.

The Problem

The problem we are trying to solve is to find the volume of a column with a square base $R=\{(x, y):|x| \leq 1,|y| \leq 1\}$ cut by the plane $z=4-x-y$ . The column has a square base with side length 2 units, and the plane cuts it at an angle. Our goal is to find the volume of the resulting solid.

Setting Up the Problem

To find the volume of the solid, we need to set up a triple integral that represents the volume of the solid. The triple integral is given by:

\iiint_R (4-x-y) \,dx\,dy\,dz

where $R$ is the region of integration, which is the square base of the column.

Converting to a Double Integral

Since the region of integration is a square, we can convert the triple integral to a double integral by using the following substitution:

x = u \cos v

y = u \sin v

z = 4 - u \cos v - u \sin v

This substitution allows us to transform the region of integration from a square to a circular region.

Evaluating the Double Integral

The double integral can be evaluated as follows:

\int_0^{2\pi} \int_0^1 (4-u \cos v - u \sin v) \,u \,du\,dv

Evaluating the inner integral, we get:

\int_0^{2\pi} \left[2u^2 - \frac{u^3}{3} \cos v - \frac{u^3}{3} \sin v\right]_0^1 \,dv

Simplifying the expression, we get:

\int_0^{2\pi} \left(2 - \frac{1}{3} \cos v - \frac{1}{3} \sin v\right) \,dv

Evaluating the integral, we get:

\left[2v - \frac{1}{3} \sin v + \frac{1}{3} \cos v\right]_0^{2\pi}

Simplifying the expression, we get:

4\pi

Conclusion

In this article, we discussed how to find the volume of a column with a square base cut by a plane using the method of integration. We set up a triple integral that represents the volume of the solid, converted it to a double integral using a substitution, and evaluated the double integral to find the volume of the solid. The final answer is $\boxed{4\pi}$ cubic units.

Discussion

The problem we solved is a classic example of a solid of revolution. The column with a square base is a solid of revolution, and the plane cuts it at an angle, creating a new solid. The method of integration is a powerful tool for finding the volume of solids of revolution. In this case, we used the concept of triple integration to find the volume of the solid, and then converted it to a double integral using a substitution. The final answer is a simple expression in terms of $\pi$ , which is a common result in problems involving solids of revolution.

Applications

The problem we solved has numerous applications in various fields, including physics, engineering, and architecture. For example, in physics, the volume of a solid can be used to calculate the mass of an object. In engineering, the volume of a solid can be used to calculate the weight of a structure. In architecture, the volume of a solid can be used to calculate the volume of a building.

Future Work

In future work, we can explore other methods for finding the volume of solids of revolution, such as the method of cylindrical shells or the method of washers. We can also explore other applications of the method of integration in various fields, such as physics, engineering, and architecture.

References

[1] "Calculus" by Michael Spivak
[2] "Calculus: Early Transcendentals" by James Stewart
[3] "Mathematics for Engineers and Scientists" by Donald R. Hill

Glossary

Solid of revolution: A solid object that is formed by rotating a region about an axis.
Triple integral: A mathematical expression that represents the volume of a solid.
Double integral: A mathematical expression that represents the volume of a solid in two dimensions.
Substitution: A technique used to simplify a mathematical expression by replacing variables with new variables.

Appendix

The following is a list of formulas and theorems used in this article:

Triple integral formula: $\iiint_R f(x,y,z) \,dx\,dy\,dz$
Double integral formula: $\iint_R f(x,y) \,dx\,dy$
Substitution formula: $x = u \cos v$
Pythagorean theorem: $a^2 + b^2 = c^2$

Note: The formulas and theorems listed above are not exhaustive, and there may be other formulas and theorems used in this article that are not listed.

Introduction

In our previous article, we discussed how to find the volume of a column with a square base cut by a plane using the method of integration. In this article, we will answer some common questions related to this topic.

Q: What is the formula for finding the volume of a column with a square base cut by a plane?

A: The formula for finding the volume of a column with a square base cut by a plane is given by:

\iiint_R (4-x-y) \,dx\,dy\,dz

where $R$ is the region of integration, which is the square base of the column.

Q: How do I set up the triple integral for finding the volume of a column with a square base cut by a plane?

A: To set up the triple integral, you need to define the region of integration, which is the square base of the column. You can do this by defining the limits of integration for $x$ , $y$ , and $z$ . For example, if the square base has side length 2 units, the limits of integration for $x$ and $y$ would be $-1 \leq x \leq 1$ and $-1 \leq y \leq 1$ , respectively.

Q: How do I convert the triple integral to a double integral?

A: To convert the triple integral to a double integral, you need to use a substitution to simplify the expression. For example, you can use the substitution:

x = u \cos v

y = u \sin v

z = 4 - u \cos v - u \sin v

This substitution allows you to transform the region of integration from a square to a circular region.

Q: How do I evaluate the double integral?

A: To evaluate the double integral, you need to integrate the expression with respect to $u$ and $v$ . You can do this by using the following formula:

\int_0^{2\pi} \int_0^1 (4-u \cos v - u \sin v) \,u \,du\,dv

Q: What is the final answer for the volume of the column with a square base cut by a plane?

A: The final answer for the volume of the column with a square base cut by a plane is $\boxed{4\pi}$ cubic units.

Q: What are some common applications of finding the volume of a column with a square base cut by a plane?

A: Some common applications of finding the volume of a column with a square base cut by a plane include:

Calculating the mass of an object
Calculating the weight of a structure
Calculating the volume of a building

Q: What are some common methods for finding the volume of a column with a square base cut by a plane?

A: Some common methods for finding the volume of a column with a square base cut by a plane include:

The method of integration
The method of cylindrical shells
The method of washers

Q: What are some common challenges when finding the volume of a column with a square base cut by a plane?

A: Some common challenges when finding the volume of a column with a square base cut by a plane include:

Defining the region of integration
Simplifying the expression using a substitution
Evaluating the double integral

Q: What are some common tools and software used for finding the volume of a column with a square base cut by a plane?

A: Some common tools and software used for finding the volume of a column with a square base cut by a plane include:

Calculators
Computer algebra systems (CAS)
Graphing software

Conclusion

In this article, we answered some common questions related to finding the volume of a column with a square base cut by a plane. We discussed the formula for finding the volume, how to set up the triple integral, how to convert the triple integral to a double integral, and how to evaluate the double integral. We also discussed some common applications, methods, challenges, and tools used for finding the volume of a column with a square base cut by a plane.

Discussion

Finding the volume of a column with a square base cut by a plane is a fundamental problem in mathematics that has numerous applications in various fields. The method of integration is a powerful tool for finding the volume of solids of revolution. In this article, we discussed some common questions related to finding the volume of a column with a square base cut by a plane and provided some common answers.

Future Work

References

[1] "Calculus" by Michael Spivak
[2] "Calculus: Early Transcendentals" by James Stewart
[3] "Mathematics for Engineers and Scientists" by Donald R. Hill

Glossary

Solid of revolution: A solid object that is formed by rotating a region about an axis.
Triple integral: A mathematical expression that represents the volume of a solid.
Double integral: A mathematical expression that represents the volume of a solid in two dimensions.
Substitution: A technique used to simplify a mathematical expression by replacing variables with new variables.

Appendix

The following is a list of formulas and theorems used in this article:

Triple integral formula: $\iiint_R f(x,y,z) \,dx\,dy\,dz$
Double integral formula: $\iint_R f(x,y) \,dx\,dy$
Substitution formula: $x = u \cos v$
Pythagorean theorem: $a^2 + b^2 = c^2$

Note: The formulas and theorems listed above are not exhaustive, and there may be other formulas and theorems used in this article that are not listed.