Understanding Double Integrals

Oct 13, 2025 by windailys.com 97 views

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Introduction

Double integrals are a fundamental concept in mathematics, particularly in the field of calculus. They are used to calculate the volume of a solid, the area of a surface, and the mass of a solid. In this article, we will evaluate the double integral $\iint_R 3x \cos(xy) \, dA$ where $R = [0,1] \times \left[0, \frac{\pi}{2}\right]$ . This will involve understanding the concept of double integrals, the properties of the integrand, and the application of Fubini's theorem.

Understanding Double Integrals

A double integral is a mathematical operation that is used to calculate the volume of a solid or the area of a surface. It is an extension of the single integral, which is used to calculate the area under a curve. The double integral is denoted by $\iint_R f(x,y) \, dA$ , where $f(x,y)$ is the integrand and $R$ is the region of integration.

The double integral can be evaluated using Fubini's theorem, which states that the double integral can be evaluated as an iterated integral. This means that the double integral can be evaluated as a single integral with respect to one variable, and then as a single integral with respect to the other variable.

Properties of the Integrand

The integrand in this problem is $3x \cos(xy)$ . This is a product of two functions, $3x$ and $\cos(xy)$ . The function $3x$ is a linear function, while the function $\cos(xy)$ is a trigonometric function.

The function $3x$ is a linear function, which means that it is a function of the form $f(x) = ax + b$ , where $a$ and $b$ are constants. In this case, $a = 3$ and $b = 0$ . The function $\cos(xy)$ is a trigonometric function, which means that it is a function of the form $f(x) = \cos(ax)$ , where $a$ is a constant. In this case, $a = y$ .

Evaluating the Double Integral

To evaluate the double integral, we will use Fubini's theorem. This means that we will evaluate the double integral as an iterated integral.

First, we will evaluate the double integral with respect to $y$ . This means that we will integrate the function $3x \cos(xy)$ with respect to $y$ , while treating $x$ as a constant.

$\int_{0}^{\frac{\pi}{2}} 3x \cos(xy) \, dy$

To evaluate this integral, we will use the substitution method. We will substitute $u = xy$ , which means that $du = y \, dx$ . This will give us:

$\int_{0}^{\frac{\pi}{2}} 3x \cos(u) \, \frac{du}{x}$

Simplifying this expression, we get:

$\int_{0}^{\frac{\pi}{2}} 3 \cos(u) \, du$

Evaluating this integral, we get:

$3 \sin(u) \, \Big|_{0}^{\frac{\pi}{2}}$

Simplifying this expression, we get:

$3 \sin\left(\frac{\pi}{2}\right) - 3 \sin(0)$

Simplifying this expression, we get:

$3 - 0$

Simplifying this expression, we get:

$3$

Evaluating the Double Integral with Respect to x

Now that we have evaluated the double integral with respect to $y$ , we can evaluate the double integral with respect to $x$ . This means that we will integrate the function $3 \sin(x)$ with respect to $x$ , while treating $y$ as a constant.

$\int_{0}^{1} 3 \sin(x) \, dx$

To evaluate this integral, we will use the substitution method. We will substitute $u = x$ , which means that $du = dx$ . This will give us:

$\int_{0}^{1} 3 \sin(u) \, du$

Evaluating this integral, we get:

$-3 \cos(u) \, \Big|_{0}^{1}$

Simplifying this expression, we get:

$-3 \cos(1) + 3 \cos(0)$

Simplifying this expression, we get:

$-3 \cos(1) + 3$

Final Answer

The final answer is the product of the two integrals that we evaluated. This means that we will multiply the two integrals together.

$3 \times (-3 \cos(1) + 3)$

Simplifying this expression, we get:

$-9 \cos(1) + 9$

Conclusion

In this article, we evaluated the double integral $\iint_R 3x \cos(xy) \, dA$ where $R = [0,1] \times \left[0, \frac{\pi}{2}\right]$ . We used Fubini's theorem to evaluate the double integral as an iterated integral. We first evaluated the double integral with respect to $y$ , and then evaluated the double integral with respect to $x$ . The final answer is the product of the two integrals that we evaluated.

References

Fubini, G. (1909). Sugli integrali d'area. Rendiconti del Circolo Matematico di Palermo, 22, 73-86.
Rudin, W. (1976). Principles of mathematical analysis. McGraw-Hill.
Spivak, M. (1965). Calculus on manifolds. Benjamin.

Future Work

In the future, we can use the techniques that we developed in this article to evaluate other double integrals. We can also use the techniques that we developed in this article to solve other problems in mathematics, such as optimization problems and differential equations.

Limitations

One limitation of this article is that we assumed that the integrand is a product of two functions. In reality, the integrand can be a more complex function. In the future, we can develop techniques to evaluate double integrals with more complex integrands.

Conclusion

In conclusion, we evaluated the double integral $\iint_R 3x \cos(xy) \, dA$ where $R = [0,1] \times \left[0, \frac{\pi}{2}\right]$ . We used Fubini's theorem to evaluate the double integral as an iterated integral. We first evaluated the double integral with respect to $y$ , and then evaluated the double integral with respect to $x$ . The final answer is the product of the two integrals that we evaluated.

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Introduction

In our previous article, we evaluated the double integral $\iint_R 3x \cos(xy) \, dA$ where $R = [0,1] \times \left[0, \frac{\pi}{2}\right]$ . We used Fubini's theorem to evaluate the double integral as an iterated integral. In this article, we will answer some common questions that readers may have about double integrals.

Q: What is a double integral?

Q: How do I evaluate a double integral?

To evaluate a double integral, you can use Fubini's theorem. This means that you will evaluate the double integral as an iterated integral. You will first evaluate the double integral with respect to one variable, and then evaluate the double integral with respect to the other variable.

Q: What is Fubini's theorem?

Fubini's theorem is a mathematical theorem that states that a double integral can be evaluated as an iterated integral. This means that you can evaluate the double integral by first evaluating the integral with respect to one variable, and then evaluating the integral with respect to the other variable.

Q: How do I apply Fubini's theorem?

To apply Fubini's theorem, you will first evaluate the double integral with respect to one variable, and then evaluate the double integral with respect to the other variable. You will use the following formula:

$\iint_R f(x,y) \, dA = \int_{a}^{b} \int_{c}^{d} f(x,y) \, dy \, dx$

Q: What is the difference between a double integral and a single integral?

A double integral is used to calculate the volume of a solid or the area of a surface, while a single integral is used to calculate the area under a curve.

Q: Can I use Fubini's theorem to evaluate any double integral?

No, you cannot use Fubini's theorem to evaluate any double integral. Fubini's theorem only applies to double integrals that are defined over a rectangular region.

Q: How do I know if a double integral is defined over a rectangular region?

A double integral is defined over a rectangular region if the region of integration is a rectangle. This means that the region of integration is a rectangle with a fixed width and height.

Q: Can I use Fubini's theorem to evaluate a double integral that is not defined over a rectangular region?

No, you cannot use Fubini's theorem to evaluate a double integral that is not defined over a rectangular region. Fubini's theorem only applies to double integrals that are defined over a rectangular region.

Q: What are some common applications of double integrals?

Double integrals have many common applications in mathematics and science. Some common applications include:

Calculating the volume of a solid
Calculating the area of a surface
Calculating the mass of a solid
Calculating the center of mass of a solid

Q: How do I use double integrals in real-world applications?

To use double integrals in real-world applications, you will need to understand the concept of double integrals and how to evaluate them. You will also need to understand the properties of the integrand and the region of integration.

Conclusion

In this article, we answered some common questions that readers may have about double integrals. We discussed the concept of double integrals, how to evaluate them, and some common applications of double integrals. We also discussed the properties of the integrand and the region of integration.

References

Fubini, G. (1909). Sugli integrali d'area. Rendiconti del Circolo Matematico di Palermo, 22, 73-86.
Rudin, W. (1976). Principles of mathematical analysis. McGraw-Hill.
Spivak, M. (1965). Calculus on manifolds. Benjamin.

Future Work

Limitations

Conclusion

In conclusion, we answered some common questions that readers may have about double integrals. We discussed the concept of double integrals, how to evaluate them, and some common applications of double integrals. We also discussed the properties of the integrand and the region of integration.