Find The Area Enclosed Between The Graph Of The Function F(x)=2+x^2, The X-axis, And The Interval [0,1].

Calculating the area bounded by a curve, the x-axis, and a given interval is a fundamental problem in calculus. This article delves into the process of finding the area under the curve of the function f(x) = 2 + x² within the interval [0, 1]. This exploration not only reinforces the understanding of definite integrals but also highlights their practical applications in various fields. Understanding how to calculate areas under curves is crucial for students and professionals alike, as it forms the basis for more advanced concepts in mathematics, physics, engineering, and economics. By mastering this technique, you'll gain a valuable tool for solving real-world problems involving rates of change, accumulation, and optimization. This article will provide a comprehensive guide to tackling this problem, ensuring clarity and understanding every step of the way.

Understanding the Problem: Area Under a Curve

To effectively find the area under a curve, it's crucial to first grasp the underlying concept. The area we seek is the region enclosed by the graph of the function f(x) = 2 + x², the x-axis, and the vertical lines x = 0 and x = 1. This area represents the accumulation of the function's values over the specified interval. Visualizing this area helps in understanding the process of calculating it. The function f(x) = 2 + x² is a parabola that opens upwards, shifted 2 units up along the y-axis. Over the interval [0, 1], the curve lies entirely above the x-axis, ensuring that the area we calculate is positive. Understanding this geometric interpretation is key to appreciating the power of integral calculus. The area under a curve can represent various physical quantities, such as distance traveled (if f(x) represents velocity) or total revenue (if f(x) represents marginal revenue). Therefore, mastering this concept is not only academically valuable but also practically significant. This article aims to guide you through the steps of calculating this area accurately and efficiently. By breaking down the process into manageable steps, we'll ensure that you understand not just the 'how' but also the 'why' behind each calculation.

The Definite Integral: The Key Tool

The definite integral serves as our primary tool for calculating the area under a curve. It provides a precise way to sum up the infinitesimal areas under the curve within the given interval. The definite integral of a function f(x) from a to b, denoted as ∫ab f(x) dx, represents the signed area between the curve f(x) and the x-axis from x = a to x = b. In our case, we need to calculate the definite integral of f(x) = 2 + x² from 0 to 1. This means we will be evaluating ∫01 (2 + x²) dx. The definite integral is a cornerstone of calculus, and its applications extend far beyond finding areas. It is used extensively in physics to calculate work done by a force, in statistics to find probabilities, and in economics to determine consumer surplus. The process of evaluating a definite integral involves finding the antiderivative of the function and then applying the Fundamental Theorem of Calculus. This theorem states that the definite integral of a function from a to b is equal to the difference between the antiderivative evaluated at b and the antiderivative evaluated at a. In the following sections, we will delve into the steps of finding the antiderivative and applying the Fundamental Theorem of Calculus to solve our problem. By understanding the theoretical foundation of the definite integral, you'll be better equipped to tackle a wide range of calculus problems.

Step-by-Step Solution: Calculating the Area

Let's embark on the step-by-step journey of calculating the area under the curve. First, we need to find the antiderivative of the function f(x) = 2 + x². Recall that the antiderivative of a function is a function whose derivative is the original function. The antiderivative of 2 is 2x, and the antiderivative of x² is (1/3)x³. Therefore, the antiderivative of f(x) = 2 + x² is F(x) = 2x + (1/3)x³ + C, where C is the constant of integration. However, when evaluating definite integrals, the constant of integration cancels out, so we can ignore it for our purposes. Next, we apply the Fundamental Theorem of Calculus. This theorem states that ∫ab f(x) dx = F(b) - F(a), where F(x) is the antiderivative of f(x). In our case, a = 0 and b = 1. So, we need to calculate F(1) - F(0). F(1) = 2(1) + (1/3)(1)³ = 2 + 1/3 = 7/3. F(0) = 2(0) + (1/3)(0)³ = 0. Therefore, the definite integral ∫01 (2 + x²) dx = F(1) - F(0) = 7/3 - 0 = 7/3. This means the area under the curve of f(x) = 2 + x² between x = 0 and x = 1 is 7/3 square units. This result provides a concrete answer to our initial problem. By breaking down the calculation into smaller, manageable steps, we have demonstrated the power and elegance of integral calculus.

Practical Implications and Applications

Understanding how to calculate the area under a curve extends far beyond the classroom. The applications of this concept are vast and varied, spanning across numerous disciplines. In physics, for instance, the area under a velocity-time curve represents the displacement of an object. Similarly, the area under a force-displacement curve represents the work done by the force. In economics, the area under a marginal cost curve can represent the total cost of production, and the area under a demand curve can represent consumer surplus. In statistics, the area under a probability density function represents the probability of an event occurring within a certain range. The ability to calculate these areas allows us to make informed decisions and predictions in various real-world scenarios. For example, engineers might use integral calculus to design structures that can withstand certain loads, economists might use it to analyze market trends, and statisticians might use it to model and predict the spread of diseases. The function f(x) = 2 + x², while seemingly simple, can represent a variety of phenomena depending on the context. For instance, it could model the cost of producing x units of a product, where the constant term 2 represents fixed costs and the x² term represents variable costs. By finding the area under this curve, we can estimate the total cost of production over a given range of output. This highlights the practical relevance of the mathematical concepts we have explored. By mastering the techniques of integral calculus, you'll gain a valuable tool for solving a wide range of real-world problems.

Conclusion: Mastering Area Calculation

In conclusion, mastering area calculation using definite integrals is a crucial skill with wide-ranging applications. We have successfully calculated the area under the curve of the function f(x) = 2 + x² within the interval [0, 1], demonstrating the power and elegance of integral calculus. By understanding the underlying concepts, including the definite integral and the Fundamental Theorem of Calculus, you can tackle a variety of similar problems. The step-by-step approach we have outlined provides a clear and concise method for solving area calculation problems. Remember, the key is to first find the antiderivative of the function and then apply the Fundamental Theorem of Calculus to evaluate the definite integral. The ability to calculate areas under curves is not just an academic exercise; it is a valuable tool for solving real-world problems in various fields. From physics and engineering to economics and statistics, the applications of integral calculus are vast and varied. By mastering this skill, you will be well-equipped to analyze and interpret data, make informed decisions, and solve complex problems. We encourage you to practice these techniques with different functions and intervals to solidify your understanding. The more you practice, the more confident and proficient you will become in applying these concepts. This article has provided a solid foundation for understanding area calculation, and we hope it has inspired you to explore the fascinating world of calculus further.