﻿ Using Area Mean Value Theorem to Solve Some Double Integrals

### Using Area Mean Value Theorem to Solve Some Double Integrals

Chii-Huei Yu, Shinn-Der Sheu

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## Using Area Mean Value Theorem to Solve Some Double Integrals

Chii-Huei Yu1,, Shinn-Der Sheu2

1Department of Management and Information, Nan Jeon University of Science and Technology, Tainan City, Taiwan

2Department of Information Technology, Nan Jeon University of Science and Technology, Tainan City, Taiwan

### Abstract

The present paper studies six types of double integrals and uses Maple for verification. These double integrals can be solved using area mean value theorem. On the other hand, some examples are used to demonstrate the calculations.

• Yu, Chii-Huei, and Shinn-Der Sheu. "Using Area Mean Value Theorem to Solve Some Double Integrals." Turkish Journal of Analysis and Number Theory 2.3 (2014): 75-79.
• Yu, C. , & Sheu, S. (2014). Using Area Mean Value Theorem to Solve Some Double Integrals. Turkish Journal of Analysis and Number Theory, 2(3), 75-79.
• Yu, Chii-Huei, and Shinn-Der Sheu. "Using Area Mean Value Theorem to Solve Some Double Integrals." Turkish Journal of Analysis and Number Theory 2, no. 3 (2014): 75-79.

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### 1. Introduction

In calculus and engineering mathematics, there are many methods to solve the integral problems including change of variables method, integration by parts method, partial fractions method, trigonometric substitution method, and so on. In this paper, we study the following six types of double integrals which are not easy to obtain their answers using the methods mentioned above.

 (1)
 (2)
 (3)
 (4)
 (5)
 (6)

where are real numbers, , and is a positive integer,

 (7)
 (8)
 (9)
 (10)
 (11)

and

 (12)

We can obtain the solutions of these double integrals using area mean value theorem; these are the major results of this paper (i.e., Theorems 1-3). Adams et al. [1], Nyblom [2], and Oster [3] provided some techniques to solve the integral problems. Yu [4-29][4], Yu and B. -H. Chen [30], and T. -J. Chen and Yu [31, 32, 33] used complex power series method, integration term by term theorem, differentiation with respect to a parameter, Parseval’s theorem, and generalized Cauchy integral formula to solve some types of integrals. In this paper, three examples are used to demonstrate the proposed calculations, and the manual calculations are verified using Maple.

### 2. Main Results

Some formulas and theorems used in this paper are introduced below.

2.1. Euler’s Formula

, where , and is any real number.

2.2. DeMoivre’s Formula

, where is any integer, and is any real number.

The following Formula 2.3 and Formula 2.4 can be found in [[34], p62]

2.3.

, where are real numbers.

2.4.

, where are real numbers.

2.5. Binomial Theorem:

, where are complex numbers, is a positive integer, for positive integers , and .

An important theorem used in this study is introduced below, which can be found in [[35], p147].

2.6. Area Mean Value Theorem

Suppose that are complex numbers, and . If is analytic in a domain which contains the closed disc , then:

Firstly, we determine the solutions of the double integrals (1) and (2).

Theorem 1 Suppose that are real numbers, , and is a positive integer. Then the double integrals

 (13)

and

 (14)

where

and

Proof Using area mean value theorem for analytic function yields:

 (15)

Let , then by Euler’s formula, DeMoivre’s formula and binomial theorem, we obtain:

 (16)

Thus,

 (17)

Using the equality of real parts of both sides of Eq. (17) yields Eq. (13) holds. Also by the equality of imaginary parts of both sides of Eq. (17), we obtain Eq. (14).

Next, the solutions of the double integrals (3) and (4) can be obtained below.

Theorem 2 If the assumptions are the same as Theorem 1, then the double integrals

 (18)

and

 (19)

where

and

Proof By area mean value theorem for analytic function , we have:

 (20)

Let , then:

 (21)

It follows that:

 (22)

By Formula 2.3 and the equality of real parts of both sides of Eq. (22), we obtain Eq. (18). Also using Formula 2.3 and the equality of imaginary parts of both sides of Eq. (22) yields Eq. (19) holds.

Finally, we solve the double integrals (5) and (6).

Theorem 3 If the assumptions are the same as Theorem 1, then the double integrals

 (23)

and

 (24)

where

and

Proof Using area mean value theorem for analytic function and Formula 2.4, we can easily obtain the desired results.

### 3. Examples

In the following, for the six types of double integrals in this study, we provide some examples and use Theorems 1-3 to determine their solutions. On the other hand, Maple is used to calculate the approximations of some double integrals and their solutions for verifying our answers.

Example 1 In Eq. (13), let , and , we obtain:

 (25)

where

 (26)

Next, we use Maple to verify the correctness of Eq. (25).

>evalf(Doubleint(r*exp(1+r*cos(theta))*cos(sqrt(3)+r*sin(theta)),r=0..2,theta=0..2*Pi),18);

>evalf(4*Pi*exp(1)*cos(sqrt(3)),18);

Also using Eq. (14) yields:

 (27)

where

 (28)

>evalf(Doubleint(r*exp(1+r*cos(theta))*sin(sqrt(3)+r*sin(theta)),r=0..2,theta=0..2*Pi),18);

>evalf(4*Pi*exp(1)*sin(sqrt(3)),18);

Example 2 In Eq. (18), if , and , then:

 (29)

where

 (30)

We also use Maple to verify the correctness of Eq. (29).

>evalf(Doubleint(r*sin(-1-2r*sin(theta)+r^2*cos(2*theta))

>evalf(-9*Pi*sin(1));

Also by Eq. (19), we have:

 (31)

where

 (32)

>evalf(Doubleint(r*cos(-1-2r*sin(theta)+r^2*cos(2*theta))

Example 3 In Eq. (23), let , and , then:

 (33)

where

 (34)

>evalf(Doubleint(r*cos(-2+r*cos(theta))*cosh(r*sin(theta

>evalf(Pi*cos(2));

On the other hand, using Eq. (24) yields:

 (35)

where

 (36)

>evalf(Doubleint(r*sin(-2+r*cos(theta))*sinh(r*sin(theta

### 4. Conclusion

In this paper, we use area mean value theorem to solve some types of double integrals. In fact, the applications of this theorem are extensive, and can be used to easily solve many difficult problems; we endeavor to conduct further studies on related applications. In addition, Maple also plays a vital assistive role in problem-solving. In the future, we will extend the research topic to other calculus and engineering mathematics problems and use Maple to verify our answers.

### References

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