Double integral

Definition

A double integral is a mathematical operation used to compute the volume under a surface in three-dimensional space, defined by a function of two variables over a specified region. This operation extends the concept of a single integral, allowing for the integration of functions across two dimensions, thereby enabling the calculation of areas, volumes, and other properties of two-variable functions.

5 Must Know Facts For Your Next Test

1. Double integrals can be evaluated over rectangular and non-rectangular regions, with different techniques applied based on the region's shape.
2. When using Fubini's theorem, the double integral can be simplified into iterated integrals, which can make calculations more manageable.
3. The change of variables theorem allows for transforming variables in double integrals, often using polar coordinates for circular regions.
4. Double integrals are essential for applications like calculating mass, moments, and centers of mass of two-dimensional objects.
5. The surface area of a function graph can be computed using double integrals by integrating the square root of the sum of the squares of the partial derivatives.

Review Questions

• How does Fubini's theorem facilitate the evaluation of double integrals?
  • Fubini's theorem allows for the evaluation of double integrals by permitting the interchange of the order of integration. This means that you can first integrate with respect to one variable and then the other, making calculations simpler depending on the function and limits involved. By breaking down a double integral into iterated integrals, it can often be easier to compute especially when one variable leads to simpler calculations than another.
• What is the significance of using polar coordinates in evaluating double integrals over circular regions?
  • Using polar coordinates simplifies the evaluation of double integrals over circular regions by transforming rectangular coordinates (x,y) into polar coordinates (r,θ). In this transformation, the limits change and the area element becomes r dr dθ. This makes it easier to compute areas and volumes under surfaces that are naturally circular or radial in symmetry, improving computational efficiency.
• Discuss how double integrals relate to calculating mass and centers of mass in two-dimensional objects.
  • Double integrals play a crucial role in calculating mass and centers of mass for two-dimensional objects by integrating density functions over a given region. The mass of an object can be found by integrating its density function across its area using a double integral. The center of mass can also be determined by evaluating moments about each axis through double integrals that take into account the object's density distribution. This relationship illustrates how double integrals connect geometry with physical properties.

Related terms

iterated integral: An iterated integral is the process of evaluating a double integral by performing two successive single integrals, one for each variable.

Fubini's theorem: Fubini's theorem states that the order of integration can be interchanged in a double integral under certain conditions, allowing for flexibility in computation.

Jacobian: The Jacobian is a determinant used in change of variables for multiple integrals, facilitating the transformation between coordinate systems.