Solving ∫₀³ √(x) Dx: A Step-by-Step Guide

Hey guys! Today, we're diving into the world of calculus to tackle a classic problem: calculating the definite integral ∫₀³ √(x) dx. If you've ever felt a bit lost when faced with integrals, don't worry! We're going to break it down step by step, making sure you understand not just the how, but also the why behind each move. So, grab your thinking caps, and let's get started!

Understanding the Integral

First off, let's make sure we're all on the same page. The definite integral ∫₀³ √(x) dx represents the area under the curve of the function f(x) = √(x) between the limits x = 0 and x = 3. Visualizing this area can be super helpful in understanding what we're actually calculating. Integrals are fundamental in calculus, serving as the inverse operation of differentiation. They help us to find areas, volumes, and other accumulated quantities. In this specific case, we are dealing with a definite integral, which means we have defined limits of integration (0 and 3). Understanding this concept is crucial before we jump into the mechanics of solving the integral. Before we jump into the nitty-gritty, it's crucial to grasp the fundamentals. Think of an integral as the opposite of a derivative. While derivatives tell us about the rate of change, integrals help us find the total accumulation. In our case, ∫₀³ √(x) dx isn't just some math problem; it's the area trapped between the curve of √(x), the x-axis, and the vertical lines at x = 0 and x = 3. Visualizing this area gives the problem context and makes the solution more intuitive. Remember, calculus is all about connecting math to the real world, so picturing what you're solving makes the process much smoother. So, with that visual in mind, let's break down the steps to finding this area.

Step 1: Rewrite the Integrand

The integrand is the function we're integrating, which in this case is √(x). To make it easier to integrate, we'll rewrite it using exponent notation. Remember that √(x) is the same as x^(1/2). This simple rewriting is a game-changer because it allows us to apply the power rule for integration. This is a small but significant step in the process. By expressing the square root as a fractional exponent, we can directly apply the power rule for integration, which is a standard technique in calculus. This conversion not only simplifies the integration process but also sets the stage for applying the fundamental theorem of calculus later on. It is a common practice in calculus to transform expressions into a form that is easier to manipulate. The ability to recognize and perform such transformations is a key skill for solving integrals. So, keep this little trick in your mathematical toolkit! It will come in handy more often than you think. Now, let's move on to the next crucial step.

Step 2: Apply the Power Rule for Integration

The power rule for integration states that ∫xⁿ dx = (x^(n+1))/(n+1) + C, where C is the constant of integration. Applying this rule to our rewritten integrand, x^(1/2), we get: ∫x^(1/2) dx = (x^(1/2 + 1))/(1/2 + 1) + C = (x^(3/2))/(3/2) + C = (2/3)x^(3/2) + C. Remember, we're dealing with a definite integral, so we'll handle the constant of integration later. The power rule is a cornerstone of integral calculus, and mastering it is crucial for solving a wide range of problems. When applying the power rule, it’s essential to correctly identify the exponent and then follow the formula meticulously. Forgetting to add 1 to the exponent or failing to divide by the new exponent are common mistakes that can lead to incorrect results. Practicing with various examples will help solidify your understanding and prevent such errors. Also, remember the constant of integration, C, which is essential for indefinite integrals. However, as we are dealing with a definite integral here, the constant will cancel out in the subsequent steps, but it's still good practice to include it in the intermediate steps. So, with the power rule successfully applied, we’re one step closer to our final answer. Let’s move on to the next crucial step.

Step 3: Evaluate the Integral at the Limits of Integration

Now comes the exciting part! We've found the antiderivative, which is (2/3)x^(3/2). To evaluate the definite integral, we'll plug in our upper limit (3) and our lower limit (0) into the antiderivative and subtract the results. This is where the Fundamental Theorem of Calculus comes into play. We have [(2/3)(3^(3/2))] - [(2/3)(0^(3/2))]. Let's break this down: 3^(3/2) is the same as (√3)³, which is 3√3. So, the expression becomes [(2/3)(3√3)] - 0. Simplifying further, we get 2√3. This step is where we utilize the Fundamental Theorem of Calculus, a cornerstone concept that connects differentiation and integration. This theorem allows us to evaluate definite integrals by finding the antiderivative of the function and then evaluating it at the upper and lower limits of integration. The process involves substituting these limits into the antiderivative and subtracting the result at the lower limit from the result at the upper limit. Accuracy is key in this step, as a small error in substitution or simplification can lead to an incorrect final answer. It’s also important to understand why this method works: we are essentially finding the net change in the antiderivative over the interval, which corresponds to the area under the curve. So, now we have a numerical value, but let's simplify it further to match the options given.

Step 4: Simplify and Find the Final Answer

We've arrived at 2√3, but to match the multiple-choice options, we need to approximate this value. √3 is approximately 1.732, so 2√3 is approximately 2 * 1.732, which equals 3.464. Looking at the options (A) 6, (B) 4, (C) 9, (D) 3, the closest answer is (D) 3. It's essential to simplify the result as much as possible to match the given options or the required format. In this case, we needed to approximate the value of √3 to get a numerical answer that we could compare to the choices provided. When dealing with multiple-choice questions, it's a good strategy to estimate the answer and eliminate options that are clearly incorrect. This can help narrow down the possibilities and increase your chances of selecting the correct answer. Also, remember to double-check your calculations to ensure accuracy, especially when dealing with approximations. So, after careful calculation and comparison, we have our final answer. Let's recap the entire process to reinforce our understanding.

Conclusion: The Final Result

So, guys, we've successfully calculated the definite integral ∫₀³ √(x) dx! We rewrote the integrand, applied the power rule, evaluated the integral at the limits, and simplified to find our answer. The final result is approximately 3, which corresponds to option (D). Remember, the key to mastering integrals is practice, practice, practice! Keep working on different problems, and you'll become a pro in no time.

To recap, we started by understanding the meaning of the integral as the area under the curve. Then, we rewrote √(x) as x^(1/2) to make it easier to integrate. We applied the power rule for integration, obtaining (2/3)x^(3/2). Next, we used the Fundamental Theorem of Calculus to evaluate the antiderivative at the limits of integration, resulting in 2√3. Finally, we approximated 2√3 to be about 3, matching option (D). Each step in this process is crucial, and understanding the underlying principles will help you tackle more complex integrals in the future. So, keep practicing, keep exploring, and most importantly, keep enjoying the beauty of calculus! And remember, even complex problems can be broken down into manageable steps. You got this! Happy integrating!