Double integrals using Fubini's Theorem and iterated integrals, with several examples. This method is how we most typically evaluate multiple integrals. We show that the idea behind Fubini's Theorem is to compute multiple integrals using areas and slicing rather than prism volumes. Geometrically this notion is different from the definition of (higher dimensional) Riemann integration, but fortunately it nearly always works!
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Multivariable Calculus Unit 4 Lecture 2: Fubini's Theorem is a powerful tool in multivariable calculus, and in this lesson, we look at how it helps us evaluate double integrals over rectangular domains. The theorem is applicable to continuous functions as well as many other functions.
The traditional method of Riemann integration involves partitioning a domain into sub-rectangles and estimating the volume under a function by summing the volumes of prisms formed over these sub-rectangles. This method, while conceptually sound, is often cumbersome and complex in practical applications.
Fubini's Theorem offers a conceptual shift from traditional Riemann integration. Instead of partitioning the domain in both the x and y directions, the theorem suggests slicing the domain in just one direction (either x or y) and summing the volumes of these slices, akin to slicing a loaf of bread.
The volume of each slice is calculated by multiplying the thickness of the slice (say, Δxi) by the area of its face. The area of the face is determined by fixing an x-coordinate x_{i^*} and integrating the function with respect to y over the interval [c, d]. This process turns a two-dimensional problem into a sequence of one-dimensional problems.
While iterated integration using Fubini's Theorem is a common practice in multivariable calculus, it conceptually differs from Riemann integration. Fubini's Theorem allows for the computation of double integrals as iterated single-variable integrals, which is a more practical approach in many cases.
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