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Chapter 7: Problem 2

Let \(I_{j}=\left[a_{j}, b_{j}\right], 1 \leq j \leq 3,\) and suppose that \(f\) is integrable on \(R=I_{1} \times I_{2} \times I_{3} .\) Prove: (a) If the integral $$ G(y, z)=\int_{a_{1}}^{b_{1}} f(x, y, z) d x $$ exists for \((y, z) \in I_{2} \times I_{3},\) then \(G\) is integrable on \(I_{2} \times I_{3}\) and $$ \int_{R} f(x, y, z) d(x, y, z)=\int_{I_{2} \times I_{3}} G(y, z) d(y, z) $$ (b) If the integral $$ H(z)=\int_{I_{1} \times I_{2}} f(x, y, z) d(x, y) $$ exists for \(z \in I_{3},\) then \(H\) is integrable on \(I_{3}\) and $$ \int_{R} f(x, y, z) d(x, y, z)=\int_{a_{3}}^{b_{3}} H(z) d z . $$

Short Answer

Expert verified
#Question# Prove the following statements related to triple integrals of a function 饾憮: (a) If the integral of 饾憮 with respect to 饾懃 exists for \((y, z) \in I_{2} \times 饾惣_{3}\), then the function G, defined as \[G(y,z)=\int_{饾憥_{1}}^{饾憦_{1}} f(x, y, z) dx,\] is integrable over the domain 饾惣2脳饾惣3 and \[\int_{饾憛} f(x, y, z) d(x, y, z) = \int_{饾惣_{2} \times 饾惣_{3}} 饾惡(饾懄,饾懅) d(饾懄,饾懅).\] (b) If the integral of 饾憮 with respect to 饾懃 and 饾懄 exists for \(z \in 饾惣_{3}\), then the function H, defined as \[饾惢(饾懅)=\int_{饾惣_{1} \times 饾惣_{2}} f(x, y, z) d(x, y),\] is integrable over 饾惣3 and \[\int_{饾憛} f(x, y, z) d(x, y, z) = \int_{饾惣_{3}} 饾惢(饾懅) 饾憫饾懅.\]

Step by step solution

01

(a) Proof of the integrability of G and equality of the triple and double integrals

To prove that G is integrable on \(I_{2} \times I_{3}\) and the given equality holds, we can follow these steps: 1. We are given that the integral of 饾憮 with respect to 饾懃 is defined on the rectangle 饾憛: $$ G(y, z) = \int_{a_{1}}^{b_{1}} f(x, y, z) dx $$ for \((y, z) \in I_{2} \times I_{3}\). 2. Since the integral of 饾憮 with respect to 饾懃 exists, we can say that the function 饾憮 is integrable over the 饾懃-axis. 3. Now we can integrate G with respect to 饾懄 and 饾懅 over the domain 饾懓2脳饾懓3: $$ \int_{I_{2} \times I_{3}} G(y, z) d(y, z) = \int_{a_{2}}^{b_{2}} \int_{a_{3}}^{b_{3}} G(y, z) dy dz $$ 4. Substitute the definition of G and apply the Fubini's theorem to change the order of integration: $$ \int_{a_{2}}^{b_{2}} \int_{a_{3}}^{b_{3}} \int_{a_{1}}^{b_{1}} f(x, y, z) dx dy dz = \int_{R} f(x, y, z) d(x, y, z) $$ 5. Therefore, G is integrable on \(I_{2} \times I_{3}\), and the given equality holds for part (a).
02

(b) Proof of the integrability of H and equality of the triple and single integrals

To prove that H is integrable on 饾懓3 and the given equality holds, we can follow these steps: 1. We are given that the integral of 饾憮 with respect to 饾懃 and 饾懄 is defined: $$ H(z) = \int_{I_{1} \times I_{2}} f(x, y, z) d(x, y) $$ for \(z \in I_{3}\). 2. Since the integral of 饾憮 with respect to 饾懃 and 饾懄 exists, we can say that the function 饾憮 is integrable over the rectangle 饾憛. 3. Now we can integrate H with respect to 饾懅 over the domain 饾懓3: $$ \int_{a_{3}}^{b_{3}} H(z) dz = \int_{a_{3}}^{b_{3}} \int_{I_{1} \times I_{2}} f(x,y,z) d(x,y) dz $$ 4. Apply Fubini's theorem to change the order of integration and obtain the triple integral: $$ \int_{a_{3}}^{b_{3}} \int_{a_{1}}^{b_{1}} \int_{a_{2}}^{b_{2}} f(x, y, z) dx dy dz = \int_{R} f(x, y, z) d(x, y, z) $$ 5. Therefore, H is integrable on 饾懓3, and the given equality holds for part (b).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integrable Functions
The term integrable functions refers to functions whose integrals, defined as a measure of the area beneath the curve over a specified interval, are finite. For example, in the given triple integral problem, the function f is considered integrable on the region R because it has a well-defined, finite triple integral over that region. The concept of integrability is essential as it ensures the function can be analyzed using integral calculus, which enables a variety of applications from calculating volumes to solving complex physics problems.

Integrability of a function, in a more technical sense, often relies on the function meeting certain criteria such as being bounded and measurable over the region of integration. In the context of multiple integrals, the integrability of a function over multi-dimensional space requires the function to be integrable over each dimension, as illustrated by the step-by-step solution provided.
Fubini's Theorem
A cornerstone of multiple integrals is Fubini鈥檚 Theorem, a principle that states that if a function is integrable on a product space, the order of integration can be changed without affecting the value of the integral. It's essentially the green light for swapping the order of integration for double and triple integrals, provided certain conditions are met.

For example, in the given triple integral problem, Fubini鈥檚 theorem allows the integral of the function f with respect to x to be computed first, followed by the integrals with respect to y and z, or in any other order. This flexibility is not only mathematically elegant but also immensely practical, often simplifying the computation of complex integrals. To use Fubini's theorem effectively, it is important to ensure the function is integrable on the respective intervals and to be aware of the theorem's limits鈥攑articularly in cases involving non-absolute integrable functions, where caution is required.
Multiple Integrals
The concept of multiple integrals extends the idea of single-variable integration into higher dimensions. For instance, while a single integral finds the area under a curve, a double integral can find the volume under a surface, and a triple integral - as in the given exercise - can find the hypervolume in a three-dimensional space.

Multiple integrals are grounded in iterated integration, working one variable at a time. This technique, highlighted in the solution steps, involves integrating a multivariable function with respect to one variable while temporarily treating the others as constants, and then repeating this process for each variable. Such an approach is instrumental in fields ranging from physics (calculating the center of mass) to probability theory (evaluating joint density functions). Understanding the geometric interpretation of multiple integrals aids in visualizing the integration process and its outcome for functions of two, three, or more variables.
Real Analysis
The field of real analysis is the rigorous study of real numbers and real-valued functions, encompassing the concepts of limits, continuity, and, critically, integration and differentiation. In the realm of real analysis, integrals are more deeply understood through the Riemann and Lebesgue theories of integration, which provide frameworks for discussing when functions are integrable and how their integrals should be interpreted.

Within this context, the exercise at hand asks us to engage with real analysis by ensuring the function f meets the necessary criteria for integration over a region R in real space. As the foundation for various advanced mathematical subjects and applications, real analysis not only deepens our comprehension of core mathematical concepts like integrals but also enables us to tackle more complex problems in mathematics and its many applications.

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Most popular questions from this chapter

Suppose that \(\mathbf{E}\) is an \(n \times n\) elementary matrix and \(\mathbf{A}\) is an arbitrary \(n \times p\) matrix. Show that \(\mathbf{E A}\) is the matrix obtained by applying to \(\mathbf{A}\) the operation by which \(\mathbf{E}\) is obtained from the \(n \times n\) identity matrix.

Let \(R=\left[a_{1}, b_{2}\right] \times\left[a_{2}, b_{2}\right] \times \cdots \times\left[a_{n}, b_{n}\right]\). Evaluate (a) \(\int_{R}\left(x_{1}+x_{2}+\cdots+x_{n}\right) d \mathbf{X}\) (b) \(\int_{R}\left(x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}\right) d \mathbf{X}\) (c) \(\int_{R} x_{1} x_{2}, \cdots x_{n} d \mathbf{X}\)

Let \(R=[0,1] \times[0,1] \times[0,1], \widetilde{R}=[0,1] \times[0,1],\) and \(f(x, y, z)=\left\\{\begin{array}{ll}2 x y+2 x z & \text { if } y \text { and } z \text { are rational, } \\ y+2 x z & \text { if } y \text { is irrational and } z \text { is rational, } \\ 2 x y+z & \text { if } y \text { is rational and } z \text { is irrational, } \\ y+z & \text { if } y \text { and } z \text { are irrational. }\end{array}\right.\) Calculate (a) \(\int_{R} f(x, y, z) d(x, y, z)\) and \(\int_{R} f(x, y, z) d(x, y, z)\) (b) \(\int_{\widetilde{R}} f(x, y, z) d(x, y)\) and \(\overline{\int_{\widetilde{R}}} f(x, y, z) d(x, y)\) (c) \(\int_{0}^{1} d y \int_{0}^{1} f(x, y, z) d x\) and \(\int_{0}^{1} d z \int_{0}^{1} d y \int_{0}^{1} f(x, y, z) d x\).

Let \(T_{\rho}=[0, \rho] \times[0, \rho], \rho>0 .\) By calculating $$ I(a)=\lim _{\rho \rightarrow \infty} \int_{T_{\rho}} e^{-x y} \sin a x d(x, y) $$ in two different ways, show that $$ \int_{0}^{\infty} \frac{\sin a x}{x} d x=\frac{\pi}{2} \quad \text { if } \quad a>0 $$

Suppose that \(f\) is continuously differentiable on a rectangle \(R .\) Show that there is a constant \(M\) such that $$ \left|\sigma-\int_{R} f(\mathbf{X}) d \mathbf{X}\right| \leq M\|P\| $$ if \(\sigma\) is any Riemann sum of \(f\) over a partition \(P\) of \(R .\)
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