Evaluate The Integral. 3 X2 + 9 Dx 0

Evaluating the Definite Integral: ∫₀³ (3x² + 9) dx

This article provides a comprehensive guide to evaluating the definite integral ∫₀³ (3x² + 9) dx. We'll cover the fundamental concepts of integration, step-by-step solution methods, and explore related applications. Understanding this seemingly simple integral forms a crucial foundation for tackling more complex integration problems in calculus and beyond.

Understanding the Problem: Definite Integrals

Before we dive into the solution, let's clarify the terminology. The expression ∫₀³ (3x² + 9) dx represents a definite integral. This differs from an indefinite integral which results in a general antiderivative plus a constant of integration (C). A definite integral, on the other hand, calculates the signed area under the curve of the function (3x² + 9) between the limits of integration, 0 and 3.

The function, f(x) = 3x² + 9, is a simple polynomial. The limits of integration, 0 and 3, specify the interval over which we're calculating the area. The 'dx' indicates that the integration is with respect to the variable x.

Step-by-Step Solution: The Power Rule of Integration

The most straightforward method to solve this definite integral is by applying the power rule of integration. The power rule states:

∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C (where n ≠ -1)

Let's apply this rule to our integral:

∫₀³ (3x² + 9) dx

First, we integrate each term separately:

1. Integrating 3x²:

Using the power rule (n=2), we get:

∫3x² dx = 3 * (x²⁺¹)/(2+1) = 3 * (x³)/3 = x³

2. Integrating 9:

Remember that 9 can be written as 9x⁰. Applying the power rule (n=0), we get:

∫9 dx = 9 * (x⁰⁺¹)/(0+1) = 9x

3. Combining the Integrals:

Combining the results from steps 1 and 2, the indefinite integral becomes:

∫(3x² + 9) dx = x³ + 9x + C

4. Evaluating the Definite Integral:

Now, we evaluate the definite integral by substituting the limits of integration (0 and 3) into the antiderivative we just found. This involves subtracting the value of the antiderivative at the lower limit from its value at the upper limit:

[x³ + 9x]₃⁰ = [(3)³ + 9(3)] - [(0)³ + 9(0)] = (27 + 27) - (0 + 0) = 54

Therefore, the definite integral ∫₀³ (3x² + 9) dx evaluates to 54. This represents the area under the curve of the function 3x² + 9 between x = 0 and x = 3.

Visualizing the Integral: Geometric Interpretation

It's helpful to visualize what we've calculated. The function f(x) = 3x² + 9 is a parabola opening upwards. The definite integral represents the area of the region bounded by the curve, the x-axis, and the vertical lines x = 0 and x = 3. While we can't easily calculate this area using simple geometric formulas (like those for rectangles or triangles), the power of integration allows us to find the exact area, which we determined to be 54 square units.

Extending the Concept: More Complex Integrals

While the example above involved a simple polynomial, the same principles apply to more complex functions. However, for more intricate functions, you might need to employ other integration techniques such as:

• Substitution (u-substitution): This technique is useful when dealing with composite functions.
• Integration by parts: Used when integrating products of functions.
• Partial fraction decomposition: Helpful for integrating rational functions.
• Trigonometric substitutions: For integrals involving trigonometric functions.
• Numerical integration methods: Approximation techniques used when analytical solutions are difficult to find. (e.g., Trapezoidal rule, Simpson's rule)

Applications of Definite Integrals

Definite integrals have a wide range of applications across various fields:

• Physics: Calculating work done by a force, displacement, velocity, and acceleration.
• Engineering: Determining areas, volumes, and centroids of shapes.
• Economics: Finding consumer surplus and producer surplus.
• Probability and Statistics: Calculating probabilities and expected values.
• Computer Graphics: Rendering curved surfaces and calculating areas.

Practice Problems

To solidify your understanding, try evaluating the following definite integrals using the power rule and other integration techniques where necessary:

1. ∫₁² (2x + 5) dx
2. ∫₀¹ (x³ - 2x) dx
3. ∫₀⁴ (√x) dx
4. ∫₁³ (1/x) dx (Hint: This involves the natural logarithm)

Conclusion: Mastering Definite Integrals

Mastering definite integration is fundamental to success in calculus and many STEM fields. This article has provided a detailed explanation of how to evaluate a simple definite integral using the power rule. Remember to practice regularly and explore different integration techniques to expand your problem-solving skills. By understanding the concepts and applying the methods correctly, you'll be well-equipped to tackle increasingly complex integration problems and unlock the powerful applications of this essential mathematical tool. Remember to always double-check your work and utilize online resources and calculators (like Wolfram Alpha) for verification, especially when dealing with more complicated integrals. The key is persistent practice and a firm grasp of the underlying principles.