Calculating the Double Integral

Oct 13, 2025 by windailys.com 103 views

Introduction

Double integrals are a fundamental concept in mathematics, particularly in calculus. They are used to calculate the area under a curve in two dimensions. In this article, we will discuss how to calculate the double integral of a given function over a specific region. We will use the function $f(x, y) = 4 - 4x$ and the triangle with vertices $(0, 0)$ , $(6, 7)$ , and $(10, 7)$ as an example.

What is a Double Integral?

A double integral is a mathematical operation that calculates the area under a curve in two dimensions. It is a way to integrate a function of two variables over a specific region. The double integral is denoted by $\iint_{D} f(x, y) \, dx \, dy$ , where $D$ is the region of integration.

Calculating the Double Integral

To calculate the double integral, we need to follow these steps:

Define the function: The function $f(x, y) = 4 - 4x$ is given.
Define the region of integration: The region of integration is the triangle with vertices $(0, 0)$ , $(6, 7)$ , and $(10, 7)$ .
Choose the order of integration: We can choose either the $x$ -order or the $y$ -order of integration. In this case, we will choose the $x$ -order.
Calculate the inner integral: The inner integral is with respect to $x$ . We need to integrate $f(x, y)$ with respect to $x$ over the interval $[0, 10]$ .
Calculate the outer integral: The outer integral is with respect to $y$ . We need to integrate the result of the inner integral with respect to $y$ over the interval $[0, 7]$ .

Step 1: Define the Function

The function $f(x, y) = 4 - 4x$ is given.

Step 2: Define the Region of Integration

The region of integration is the triangle with vertices $(0, 0)$ , $(6, 7)$ , and $(10, 7)$ .

Step 3: Choose the Order of Integration

We can choose either the $x$ -order or the $y$ -order of integration. In this case, we will choose the $x$ -order.

Step 4: Calculate the Inner Integral

The inner integral is with respect to $x$ . We need to integrate $f(x, y)$ with respect to $x$ over the interval $[0, 10]$ .

\int_{0}^{10} (4 - 4x) \, dx

To evaluate this integral, we can use the power rule of integration, which states that $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ .

\int_{0}^{10} (4 - 4x) \, dx = \left[4x - 2x^2\right]_{0}^{10}

Evaluating this expression at the limits of integration, we get:

\left[4x - 2x^2\right]_{0}^{10} = (40 - 200) - (0 - 0) = -160

Step 5: Calculate the Outer Integral

The outer integral is with respect to $y$ . We need to integrate the result of the inner integral with respect to $y$ over the interval $[0, 7]$ .

\int_{0}^{7} -160 \, dy

To evaluate this integral, we can use the constant multiple rule of integration, which states that $\int k \, dy = ky + C$ .

\int_{0}^{7} -160 \, dy = -160y + C

Evaluating this expression at the limits of integration, we get:

-160y + C = (-1120) - (0) = -1120

Conclusion

In this article, we discussed how to calculate the double integral of a given function over a specific region. We used the function $f(x, y) = 4 - 4x$ and the triangle with vertices $(0, 0)$ , $(6, 7)$ , and $(10, 7)$ as an example. We followed the steps of defining the function, defining the region of integration, choosing the order of integration, calculating the inner integral, and calculating the outer integral. The final answer is $\boxed{-1120}$ .

References

[1] Stewart, J. (2016). Calculus: Early Transcendentals. Cengage Learning.
[2] Anton, H. (2016). Calculus: Early Transcendentals. John Wiley & Sons.

Mathematical Formulas

Power Rule of Integration: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$
Constant Multiple Rule of Integration: $\int k \, dy = ky + C$

Code

import numpy as np
def double_integral(f, x_min, x_max, y_min, y_max):
# Define the function
def f(x, y):
return 4 - 4*x
# Calculate the inner integral
inner_integral = np.trapz([f(x, y) for x in np.linspace(x_min, x_max, 100)], dx=0.1)

# Calculate the outer integral
outer_integral = np.trapz([inner_integral for y in np.linspace(y_min, y_max, 100)], dx=0.1)

return outer_integral

f = lambda x, y: 4 - 4*x
x_min, x_max = 0, 10
y_min, y_max = 0, 7
result = double_integral(f, x_min, x_max, y_min, y_max)
print(result)

Q&A: Double Integrals

Q: What is a double integral?

A: A double integral is a mathematical operation that calculates the area under a curve in two dimensions. It is a way to integrate a function of two variables over a specific region.

Q: How do I calculate a double integral?

A: To calculate a double integral, you need to follow these steps:

Define the function: The function you want to integrate.
Define the region of integration: The region over which you want to integrate the function.
Choose the order of integration: You can choose either the $x$ -order or the $y$ -order of integration.
Calculate the inner integral: The inner integral is with respect to $x$ or $y$ .
Calculate the outer integral: The outer integral is with respect to the other variable.

Q: What is the difference between the $x$ -order and $y$ -order of integration?

A: The $x$ -order of integration means that you integrate with respect to $x$ first, and then with respect to $y$ . The $y$ -order of integration means that you integrate with respect to $y$ first, and then with respect to $x$ .

Q: How do I choose the order of integration?

A: You can choose either the $x$ -order or the $y$ -order of integration. The choice of order depends on the region of integration and the function you are integrating.

Q: What is the power rule of integration?

A: The power rule of integration states that $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ .

Q: What is the constant multiple rule of integration?

A: The constant multiple rule of integration states that $\int k \, dy = ky + C$ .

Q: How do I use the power rule and constant multiple rule of integration?

A: To use the power rule and constant multiple rule of integration, you need to identify the power of $x$ and the constant multiple in the function you are integrating. Then, you can apply the corresponding rule to integrate the function.

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

A: A single integral is a mathematical operation that calculates the area under a curve in one dimension. A double integral is a mathematical operation that calculates the area under a curve in two dimensions.

Q: How do I use a double integral to calculate the area under a curve?

A: To use a double integral to calculate the area under a curve, you need to define the function and the region of integration. Then, you can calculate the double integral using the steps outlined above.

Q: What are some common applications of double integrals?

A: Double integrals have many applications in mathematics, physics, and engineering. Some common applications include:

Calculating the area under a curve in two dimensions
Calculating the volume of a solid
Calculating the mass of a solid
Calculating the center of mass of a solid

Q: How do I use a double integral to calculate the volume of a solid?

A: To use a double integral to calculate the volume of a solid, you need to define the function and the region of integration. Then, you can calculate the double integral using the steps outlined above.

Q: What are some common mistakes to avoid when calculating double integrals?

A: Some common mistakes to avoid when calculating double integrals include:

Failing to define the function and the region of integration
Choosing the wrong order of integration
Failing to calculate the inner integral correctly
Failing to calculate the outer integral correctly

Q: How do I check my work when calculating double integrals?

A: To check your work when calculating double integrals, you need to:

Verify that you have defined the function and the region of integration correctly
Verify that you have chosen the correct order of integration
Verify that you have calculated the inner integral correctly
Verify that you have calculated the outer integral correctly

Conclusion

In this article, we discussed the concept of double integrals and how to calculate them. We also answered some common questions about double integrals and provided some tips for avoiding common mistakes. We hope that this article has been helpful in understanding double integrals and how to use them to calculate the area under a curve in two dimensions.

References

[1] Stewart, J. (2016). Calculus: Early Transcendentals. Cengage Learning.
[2] Anton, H. (2016). Calculus: Early Transcendentals. John Wiley & Sons.

Mathematical Formulas

Power Rule of Integration: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$
Constant Multiple Rule of Integration: $\int k \, dy = ky + C$

Code

import numpy as np
def double_integral(f, x_min, x_max, y_min, y_max):
# Define the function
def f(x, y):
return 4 - 4*x
# Calculate the inner integral
inner_integral = np.trapz([f(x, y) for x in np.linspace(x_min, x_max, 100)], dx=0.1)

# Calculate the outer integral
outer_integral = np.trapz([inner_integral for y in np.linspace(y_min, y_max, 100)], dx=0.1)

return outer_integral

f = lambda x, y: 4 - 4*x
x_min, x_max = 0, 10
y_min, y_max = 0, 7
result = double_integral(f, x_min, x_max, y_min, y_max)
print(result)

This code uses the numpy library to calculate the double integral using the trapz function. The double_integral function takes in the function f, the limits of integration x_min, x_max, y_min, and y_max, and returns the result of the double integral. The code defines the function f, the region of integration, and calculates the double integral using the double_integral function. The result is printed to the console.