Fubini's theorem

Definition

Fubini's theorem is a fundamental principle in calculus that allows the evaluation of double integrals by iteratively integrating with respect to one variable at a time. This theorem establishes that if a function is continuous on a rectangular region, then the double integral can be computed as an iterated integral, making it possible to switch the order of integration without changing the value of the integral.

5 Must Know Facts For Your Next Test

1. Fubini's theorem applies primarily to functions that are continuous or piecewise continuous over a rectangular region.
2. It allows for switching the order of integration in double integrals without affecting the final result, simplifying calculations.
3. For Fubini's theorem to hold, the integrand must be integrable over the specified region, ensuring no undefined behavior occurs.
4. The theorem extends to triple integrals as well, enabling iterated integration for three-dimensional volumes.
5. In polar coordinates, Fubini's theorem still holds, allowing for integration over circular regions by converting Cartesian coordinates.

Review Questions

• How does Fubini's theorem simplify the process of evaluating double integrals?
  • Fubini's theorem simplifies evaluating double integrals by allowing us to break them down into iterated integrals. By integrating one variable at a time, we can tackle complex integrals more easily. This method also enables us to switch the order of integration when necessary, which can lead to simpler calculations depending on the function and the limits of integration.
• What conditions must be satisfied for Fubini's theorem to apply to a given double integral?
  • For Fubini's theorem to apply, the function being integrated must be continuous or piecewise continuous over the region of integration. Additionally, the function should be integrable within that region. When these conditions are met, we can confidently switch the order of integration or use iterated integrals without affecting the outcome.
• Evaluate a double integral using Fubini's theorem and explain how changing the order of integration affects your approach.
  • To evaluate a double integral using Fubini's theorem, start by determining the bounds for both variables. For instance, if you have $$ ext{I} = \\int_{a}^{b} \\int_{c}^{d} f(x,y) dy \, dx$$, you can first integrate with respect to $y$ and then $x$. If we change the order to integrate $x$ first, $$ ext{I} = \\int_{c}^{d} \\int_{a}^{b} f(x,y) dx \, dy$$, it's essential to ensure that you adjust your limits accordingly. Changing the order might simplify computations due to the structure of $f(x,y)$ or how it behaves in different directions across the region.

Related terms

Iterated Integral:

An iterated integral is a way of expressing a multiple integral as a repeated integral, usually calculated by integrating one variable at a time.

Region of Integration: The region of integration defines the area over which a double or triple integral is calculated, and can be either rectangular or more complex shapes.

Absolute Convergence: A property that ensures the interchange of limits and integration is valid, requiring that the integral of the absolute value of the function converges.