Evaluate The Integral. 7 X2 + 49 Dx 0

Evaluating the Definite Integral: ∫₀⁷ (7x² + 49) dx

This article provides a comprehensive guide on how to evaluate the definite integral ∫₀⁷ (7x² + 49) dx. We'll break down the process step-by-step, covering the fundamental concepts of integration, the power rule, and the application of definite integrals. Furthermore, we'll explore the practical significance of definite integrals and their applications in various fields.

Understanding Definite Integrals

A definite integral calculates the net signed area between a function's curve and the x-axis over a specified interval. Unlike indefinite integrals, which result in a family of functions, definite integrals yield a numerical value. The notation ∫ₐᵇ f(x) dx represents the definite integral of the function f(x) from x = a to x = b. The values 'a' and 'b' are called the limits of integration.

In our case, we have:

∫₀⁷ (7x² + 49) dx

Here, f(x) = 7x² + 49, a = 0, and b = 7. This means we need to find the net signed area under the curve of the function 7x² + 49 from x = 0 to x = 7.

The Power Rule of Integration

To solve this definite integral, we'll utilize the power rule of integration. The power rule states that the integral of xⁿ is (xⁿ⁺¹)/(n+1) + C, where n is any real number except -1, and C is the constant of integration. For definite integrals, the constant of integration cancels out, so we don't need to worry about it.

Let's apply the power rule to each term in our integrand (7x² + 49):

• The integral of 7x²: The power is 2, so applying the power rule, we get (7x³)/3.

• The integral of 49: This is a constant function. The integral of a constant 'k' is kx. Therefore, the integral of 49 is 49x.

Therefore, the indefinite integral of (7x² + 49) is (7x³/3) + 49x.

Evaluating the Definite Integral

Now that we have the indefinite integral, we can evaluate the definite integral using the Fundamental Theorem of Calculus. This theorem states that the definite integral of a function f(x) from a to b is equal to F(b) - F(a), where F(x) is the antiderivative of f(x).

In our case, F(x) = (7x³/3) + 49x. So, we need to calculate F(7) - F(0):

F(7) = (7(7)³/3) + 49(7) = (7(343)/3) + 343 = (2401/3) + 343 = 800.333... + 343 = 1143.333...

F(0) = (7(0)³/3) + 49(0) = 0

Therefore, the definite integral is:

∫₀⁷ (7x² + 49) dx = F(7) - F(0) = 1143.333... - 0 = 1143.333...

We can express this as a fraction: 3430/3

Significance and Applications of Definite Integrals

Definite integrals have far-reaching applications across various scientific and engineering disciplines. Some notable applications include:

• Calculating Area: As mentioned earlier, the most fundamental application is calculating the area under a curve. This is crucial in geometry, physics, and other fields where determining areas of irregular shapes is necessary.

• Finding the Average Value of a Function: Definite integrals can be used to determine the average value of a function over a given interval. This is useful in statistics, signal processing, and other areas where average values are important.

• Determining Work Done: In physics, definite integrals are used to calculate the work done by a force over a given distance. This is particularly important in mechanics and thermodynamics.

• Computing Volumes: Definite integrals are essential in calculating the volumes of solids of revolution. This is a key concept in calculus and its applications in engineering and design.

• Probability and Statistics: Definite integrals play a central role in probability and statistics. The area under a probability density function represents probability, and definite integrals are used to calculate probabilities.

Advanced Techniques for Evaluating Integrals (Beyond the Scope of this Problem)

While the power rule sufficed for this problem, many other integration techniques exist for more complex integrals, including:

• Integration by Substitution (u-substitution): This method simplifies integrals by substituting a new variable for a part of the integrand.

• Integration by Parts: This technique is used to integrate products of functions.

• Trigonometric Substitution: This involves substituting trigonometric functions to simplify integrals involving certain types of expressions.

• Partial Fraction Decomposition: This method is used to integrate rational functions (ratios of polynomials).

Conclusion

Evaluating the definite integral ∫₀⁷ (7x² + 49) dx involves applying the power rule of integration and the Fundamental Theorem of Calculus. The result, 3430/3 or approximately 1143.33, represents the net signed area under the curve of the function 7x² + 49 from x = 0 to x = 7. Definite integrals are powerful tools with extensive applications in various fields, highlighting their importance in mathematics and its applications to the real world. This problem serves as a foundational example of how to approach and solve definite integrals, building a solid base for understanding more complex integration problems in the future. Remember to practice regularly and explore different integration techniques to enhance your understanding and problem-solving skills.