Double Integrals

Definition

Double integrals are a type of multiple integral used to calculate the volume of a three-dimensional object or the area of a two-dimensional region. They involve integrating a function over a two-dimensional domain, such as a region in the xy-plane.

5 Must Know Facts For Your Next Test

1. Double integrals can be used to calculate the area of a two-dimensional region in the xy-plane, such as the region bounded by a closed curve.
2. The order of integration in a double integral can be reversed, with the integral with respect to x being evaluated first or the integral with respect to y being evaluated first.
3. When working in polar coordinates, the double integral can be transformed into an integral over a region in the $r$-$\theta$ plane, with the change of variables $x = r\cos\theta$ and $y = r\sin\theta$.
4. The area of a region in the xy-plane bounded by the curves $y = f(x)$ and $y = g(x)$ can be calculated using a double integral with the limits of integration $a \leq x \leq b$ and $g(x) \leq y \leq f(x)$.
5. Double integrals can also be used to calculate the arc length of a curve in the xy-plane, by integrating the square root of $1 + (\frac{dy}{dx})^2$ over the curve.

Review Questions

• Explain how double integrals can be used to calculate the area of a two-dimensional region in the xy-plane.
  • Double integrals can be used to calculate the area of a two-dimensional region in the xy-plane by integrating a function over the region. The double integral takes the form $\iint_R f(x, y) \, dA$, where $R$ is the region of integration and $f(x, y)$ is the function being integrated. The limits of integration are set up to cover the entire region, with the inner integral typically taken with respect to y and the outer integral with respect to x. By evaluating this double integral, the area of the region can be determined.
• Describe how the change of variables from Cartesian coordinates to polar coordinates affects the evaluation of a double integral.
  • When working with double integrals in polar coordinates, the change of variables from Cartesian coordinates $(x, y)$ to polar coordinates $(r, \theta)$ can simplify the integration process. The transformation is made using the equations $x = r\cos\theta$ and $y = r\sin\theta$, which also requires a change in the differential elements from $dA = dx \, dy$ to $dA = r \, dr \, d\theta$. This allows the double integral to be rewritten in polar form as $\iint_R f(r, \theta) \, r \, dr \, d\theta$, where the limits of integration are now defined in the $r$-$\theta$ plane instead of the $x$-$y$ plane.
• Analyze how the order of integration in a double integral can be reversed, and explain the significance of this property.
  • The order of integration in a double integral can be reversed, meaning the integral with respect to x can be evaluated first or the integral with respect to y can be evaluated first. This property is expressed mathematically as $\iint_R f(x, y) \, dA = \iint_R f(x, y) \, dy \, dx = \iint_R f(x, y) \, dx \, dy$. The ability to reverse the order of integration is significant because it allows the double integral to be evaluated in the way that is most convenient or efficient for the given problem. This flexibility can simplify the integration process and make certain double integrals easier to compute.

Related terms

Multiple Integrals: Multiple integrals are integrals with more than one variable, such as double integrals and triple integrals, used to calculate volumes, areas, and other multidimensional quantities.

Iterated Integrals: Iterated integrals are a way of evaluating multiple integrals by breaking them down into a sequence of single integrals, which can be computed one at a time.

Change of Variables: Change of variables is a technique used to transform the variables in a multiple integral, often to simplify the integration process, such as when working in polar coordinates.