Evaluating $-2 \int_3^0(4-x)^{\frac{1}{2}} Dx + \int_3^0 2 Dx$ A Step-by-Step Solution

Hey guys! Let's dive into this interesting definite integral problem. We're going to break it down step-by-step, so you can understand exactly how to solve it. Our main goal here is to evaluate the integral expression: $-2 \int_3^0(4-x){\frac{1}{2}} dx + \int_3^0 2 dx$ This might look a bit intimidating at first, but don't worry, we'll tackle it together using some fundamental calculus techniques. We'll focus on u-substitution and basic integration rules to get to the final answer. So, grab your pencils and let's get started!

Understanding the Problem

Before we jump into solving, it’s crucial to understand what the problem is asking. We are given a definite integral expression that combines two separate integrals. The first integral involves a composite function, $(4-x)^{\frac{1}{2}}$, which means we'll likely need to use u-substitution. The second integral is simpler, involving a constant function, 2. Both integrals have the same limits of integration, from 3 to 0. This is important because it allows us to potentially combine the integrals later if needed, but let’s take it one step at a time. Essentially, we're calculating the signed area under the curves defined by these functions between the given limits. Definite integrals are powerful tools in calculus, allowing us to find areas, volumes, and much more. Understanding the setup is the first step to conquering any integral problem. We need to be meticulous with each step, ensuring we handle the negative signs, fractional exponents, and limits of integration correctly. Remember, the order of the limits matters; integrating from 3 to 0 is the opposite of integrating from 0 to 3, and this affects the sign of the result. So, let's keep this in mind as we move forward. Our aim is to break down the complex integral into manageable parts, solve each part individually, and then combine the results. This approach will help us minimize errors and ensure we arrive at the correct solution. With a clear understanding of the problem, we're now ready to move on to the next phase: solving the integrals.

Solving the First Integral: $-2 \int_3^0(4-x)^{\frac{1}{2}} dx$

The first integral we need to tackle is $-2 \int_3^0(4-x)^{\frac{1}{2}} dx$. This one looks a bit tricky because of the composite function $(4-x)^{\frac{1}{2}}$. But don't fret! We can use a technique called u-substitution to simplify it.

Step 1: U-Substitution

The key to u-substitution is identifying a suitable 'u'. In this case, let's choose $u = 4 - x$. This makes the expression inside the parentheses much simpler. Now, we need to find $du$, which is the derivative of $u$ with respect to $x$. So, $du = -dx$. But wait, we have $dx$ in our integral, not $-dx$. No problem! We can simply multiply both sides of the equation by -1 to get $-du = dx$. This is a crucial step, guys! Now we have everything we need to substitute. Our integral now looks like this: $-2 \int (u)^{\frac{1}{2}} (-du)$. Notice the negative sign from the $du$! We can pull that out of the integral, making it $2 \int u^{\frac{1}{2}} du$. See? It's already looking simpler. But hold on, we're not quite done with the substitution. We need to change the limits of integration too! Remember, our original limits were in terms of $x$, but now we're integrating with respect to $u$.

Step 2: Changing the Limits of Integration

This is a step that's easy to overlook, but super important! Our original limits were $x = 3$ and $x = 0$. We need to convert these to $u$ values using our substitution $u = 4 - x$. When $x = 3$, $u = 4 - 3 = 1$. When $x = 0$, $u = 4 - 0 = 4$. So, our new limits of integration are $u = 1$ and $u = 4$. This means our integral now looks like this: $2 \int_1^4 u^{\frac{1}{2}} du$. Much better, right? We've successfully transformed the integral into a much simpler form. Now we can finally integrate!

Step 3: Integrating and Evaluating

Now for the fun part! We need to find the antiderivative of $u^{\frac{1}{2}}$. Remember the power rule for integration? We add 1 to the exponent and divide by the new exponent. So, the antiderivative of $u^{\frac{1}{2}}$ is $\frac{u^{\frac{3}{2}}}{\frac{3}{2}}$, which simplifies to $\frac{2}{3}u^{\frac{3}{2}}$. Don't forget the constant 2 that's out in front! So we have $2 * \frac{2}{3}u^{\frac{3}{2}} = \frac{4}{3}u^{\frac{3}{2}}$. Now we need to evaluate this from $u = 1$ to $u = 4$. This means we plug in the upper limit (4) and subtract the result of plugging in the lower limit (1). So, we have $\frac{4}{3}(4)^{\frac{3}{2}} - \frac{4}{3}(1)^{\frac{3}{2}}$. Let's simplify this. $(4)^{\frac{3}{2}}$ is the same as $(\sqrt{4})^3 = 2^3 = 8$. And $(1)^{\frac{3}{2}}$ is just 1. So we have $\frac{4}{3}(8) - \frac{4}{3}(1) = \frac{32}{3} - \frac{4}{3} = \frac{28}{3}$. Woohoo! We've solved the first integral. That wasn't so bad, right? U-substitution is a powerful tool, and mastering it will help you conquer many integral problems. Now, let's move on to the second integral.

Solving the Second Integral: $\int_3^0 2 dx$

The second integral, $\int_3^0 2 dx$, is much simpler than the first one. This integral involves a constant function, which makes it a straightforward application of basic integration rules. Let's break it down step-by-step to ensure we get the correct result.

Step 1: Finding the Antiderivative

The first step in solving any integral is to find the antiderivative of the function inside the integral. In this case, our function is simply the constant 2. The antiderivative of a constant $k$ is $kx$, where $x$ is the variable of integration. Therefore, the antiderivative of 2 with respect to $x$ is $2x$. This is a fundamental rule of integration, and it's essential to have it memorized. Integrating constants is a basic skill in calculus, and it's used extensively in more complex problems. So, make sure you're comfortable with this concept.

Step 2: Evaluating the Definite Integral

Now that we have the antiderivative, $2x$, we need to evaluate it over the given limits of integration, which are from 3 to 0. This means we will plug in the upper limit (0) and subtract the result of plugging in the lower limit (3). Remember, the order of the limits is crucial. Integrating from 3 to 0 is not the same as integrating from 0 to 3; the sign of the result will be different. So, let's carefully plug in the limits. We have $2(0) - 2(3) = 0 - 6 = -6$. Notice the negative sign! This is because we are integrating in the reverse direction (from 3 to 0). If we had integrated from 0 to 3, the result would have been positive 6. Definite integrals represent the signed area under a curve, and the sign indicates whether the area is above or below the x-axis. In this case, the negative sign tells us that the area is below the x-axis. So, we've successfully evaluated the second integral. It was much quicker than the first one, thanks to the simplicity of the constant function. Now, we have the results of both integrals, and it's time to combine them to get the final answer.

Combining the Results

Alright, we've done the heavy lifting! We've successfully evaluated both integrals separately. Now it's time to put the pieces together and find the final answer. Remember, our original problem was: $-2 \int_3^0(4-x)\frac{1}{2}} dx + \int_3^0 2 dx$ We found that the first integral, $-2 \int_3^0(4-x)^{\frac{1}{2}} dx$, equals $\frac{28}{3}$. And the second integral, $\int_3^0 2 dx$, equals -6. So now we just need to add these two results together $\frac{28{3} + (-6)$. To add these, we need a common denominator. We can rewrite -6 as $-\frac{18}{3}$. Now we have $\frac{28}{3} - \frac{18}{3}$. Subtracting the numerators, we get $\frac{28 - 18}{3} = \frac{10}{3}$. And there you have it! The final answer is $\frac{10}{3}$. This is the value of the definite integral expression. We've successfully navigated through u-substitution, changed limits of integration, applied basic integration rules, and combined the results. This is a fantastic accomplishment! Remember, practice makes perfect. The more you work through these types of problems, the more comfortable you'll become with the techniques. So, keep practicing, keep asking questions, and keep exploring the fascinating world of calculus. We've shown here that even seemingly complex integrals can be tackled systematically, one step at a time.

Final Answer

So, to recap, we started with the integral expression $-2 \int_3^0(4-x)\frac{1}{2}} dx + \int_3^0 2 dx$. We broke it down into two separate integrals, used u-substitution to solve the first one, and applied basic integration rules to solve the second one. We carefully changed the limits of integration and combined the results. After all the steps, we arrived at the final answer $\frac{10{3}$. This result represents the signed area under the curves defined by the given functions, taking into account the specified limits of integration. We've demonstrated a comprehensive approach to solving definite integrals, highlighting the importance of u-substitution, changing limits, and combining results. Understanding these techniques is crucial for mastering calculus and its applications. Remember, calculus is a journey, and every problem you solve adds to your understanding. So, celebrate your success in solving this integral, and get ready for the next challenge! We've shown that with careful attention to detail and a systematic approach, even complex problems can be broken down and solved. Keep practicing, keep learning, and keep pushing your mathematical boundaries. The world of calculus is vast and fascinating, and you've just taken another step forward in your journey. Great job, guys!