key term - Double Integral

Definition

A double integral is a type of multiple integral used to calculate the volume of a three-dimensional object or the mass of a two-dimensional object. It represents the integration of a function over a two-dimensional region in the xy-plane.

5 Must Know Facts For Your Next Test

1. The double integral is denoted by the symbol $\iint$ and is evaluated over a two-dimensional region in the xy-plane.
2. Double integrals can be used to calculate the volume of a three-dimensional object or the mass of a two-dimensional object when the density function is known.
3. The order of integration in a double integral can be reversed, as long as the limits of integration are adjusted accordingly.
4. The change of variables formula for double integrals involves the Jacobian of the transformation between coordinate systems.
5. Double integrals are a fundamental tool in the study of moments and centers of mass, as they allow for the calculation of these important quantities.

Review Questions

• Explain how a double integral can be used to calculate the volume of a three-dimensional object.
  • To calculate the volume of a three-dimensional object using a double integral, the object is first projected onto the xy-plane, forming a two-dimensional region. The double integral is then evaluated over this region, with the limits of integration defining the boundaries of the object. The result of the double integral represents the volume of the three-dimensional object.
• Describe how the order of integration in a double integral can be reversed, and discuss the implications of this reversal.
  • The order of integration in a double integral can be reversed, as long as the limits of integration are adjusted accordingly. This is known as the Fubini theorem. Reversing the order of integration can sometimes simplify the evaluation of the integral, as the new limits of integration may be easier to work with. However, the choice of integration order can also affect the interpretation of the integral, such as when calculating moments and centers of mass.
• Analyze the role of the Jacobian in the change of variables formula for double integrals, and explain how it is used to transform the integral between different coordinate systems.
  • The Jacobian plays a crucial role in the change of variables formula for double integrals. The Jacobian is the determinant of the matrix of partial derivatives of the transformation between coordinate systems. When performing a change of variables in a double integral, the Jacobian is used to adjust the integrand and the limits of integration, ensuring that the integral correctly represents the same region in the new coordinate system. This transformation is essential when working with moments and centers of mass, as it allows the integral to be evaluated in the most convenient coordinate system.

Related terms

Multiple Integral: A multiple integral is an integral where the domain of integration is a region in two or more dimensions, rather than an interval on the real line.

Iterated Integral: An iterated integral is a multiple integral where the integration is performed one variable at a time, with the inner integral depending on the outer variable.

Jacobian: The Jacobian is the determinant of the matrix of partial derivatives of a vector-valued function, and it is used to transform double integrals between different coordinate systems.