įor a rectangular box, the order of integration does not make any significant difference in the level of difficulty in computation. ∫ e f ∫ c d ∫ a b f ( x, y, z ) d x d y d z = ∫ e f ( ∫ c d ( ∫ a b f ( x, y, z ) d x ) d y ) d z = ∫ c d ( ∫ e f ( ∫ a b f ( x, y, z ) d x ) d z ) d y = ∫ a b ( ∫ e f ( ∫ c d f ( x, y, z ) d y ) d z ) d x = ∫ e f ( ∫ a b ( ∫ c d f ( x, y, z ) d y ) d x ) d z = ∫ c e ( ∫ a b ( ∫ e f f ( x, y, z ) d z ) d x ) d y = ∫ a b ( ∫ c e ( ∫ e f f ( x, y, z ) d z ) d y ) d x. Just as in the case of the double integral, we can have an iterated triple integral, and consequently, a version of Fubini’s thereom for triple integrals exists.
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Now that we have developed the concept of the triple integral, we need to know how to compute it. Just as the double integral has many practical applications, the triple integral also has many applications, which we discuss in later sections. The sample point ( x i j k *, y i j k *, z i j k * ) ( x i j k *, y i j k *, z i j k * ) can be any point in the rectangular sub-box B i j k B i j k and all the properties of a double integral apply to a triple integral. However, continuity is sufficient but not necessary in other words, f f is bounded on B B and continuous except possibly on the boundary of B. Therefore, we will use continuous functions for our examples. Also, the triple integral exists if f ( x, y, z ) f ( x, y, z ) is continuous on B. When the triple integral exists on B, B, the function f ( x, y, z ) f ( x, y, z ) is said to be integrable on B. Then the rectangular box B B is subdivided into l m n l m n subboxes B i j k = × ×, B i j k = × ×, as shown in Figure 5.40. We divide the interval into l l subintervals of equal length Δ x = x i − x i − 1 l, Δ x = x i − x i − 1 l, divide the interval into m m subintervals of equal length Δ y = y j − y j − 1 m, Δ y = y j − y j − 1 m, and divide the interval into n n subintervals of equal length Δ z = z k − z k − 1 n. We follow a similar procedure to what we did in Double Integrals over Rectangular Regions. We can define a rectangular box B B in ℝ 3 ℝ 3 as B =. Later in this section we extend the definition to more general regions in ℝ 3. In this section we define the triple integral of a function f ( x, y, z ) f ( x, y, z ) of three variables over a rectangular solid box in space, ℝ 3. In Double Integrals over Rectangular Regions, we discussed the double integral of a function f ( x, y ) f ( x, y ) of two variables over a rectangular region in the plane.
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