Calculating Area Under Curve Using The Fundamental Theorem Of Calculus
Hey guys! Today, we're diving into a super important concept in calculus: finding the area under a curve. Specifically, we're going to use the Fundamental Theorem of Calculus to figure out the area under the curve of the function $f(x) = 3x + 13$ between the points $x = 14$ and $x = 17$. Sounds exciting, right? Let's break it down step-by-step so everyone can follow along. Understanding the area under the curve is not just some abstract math problem; it has tons of real-world applications in physics, engineering, economics, and even statistics. So, buckle up, and let's get started!
Understanding the Area Under a Curve
Before we jump into the nitty-gritty calculations, let's take a moment to really understand what we're trying to find. Imagine you've got a graph of our function, $f(x) = 3x + 13$. It's a straight line, sloping upwards, because it's a linear function. Now, picture the region trapped between this line, the x-axis, and the vertical lines at $x = 14$ and $x = 17$. The area under the curve is simply the area of this region. Visually, it's like coloring in that space and then figuring out how many square units you've colored. This area can represent various things depending on the context. For example, if $f(x)$ represented the velocity of an object over time, then the area under the curve would represent the displacement of the object during that time interval. Or, if $f(x)$ represented a rate of production, the area could represent the total amount produced. It's all about connecting the math to the real world, which is what makes calculus so powerful. Calculating this area might seem tricky at first, especially for more complex curves. You might think about breaking it up into tiny rectangles and adding up their areas. That's actually the basic idea behind integration! The Fundamental Theorem of Calculus provides a much more efficient and elegant way to do this. It links the concept of finding areas to the concept of finding antiderivatives, which is a game-changer. So, instead of summing up a gazillion tiny rectangles, we can use a simple formula based on antiderivatives. Stick with me, and you'll see how it works.
The Fundamental Theorem of Calculus: A Quick Recap
Okay, let's talk about the Fundamental Theorem of Calculus (FTC). This theorem is the superhero of calculus, the linchpin that connects differentiation and integration. It comes in two parts, but for our purpose of finding the area under the curve, we're primarily interested in the second part. This part, often called the Evaluation Theorem, gives us a straightforward method to calculate definite integrals. In plain English, the Evaluation Theorem states that if you want to find the definite integral of a function $f(x)$ from $a$ to $b$, you just need to find an antiderivative $F(x)$ of $f(x)$, and then calculate $F(b) - F(a)$. That's it! It sounds simple, but the implications are profound. An antiderivative is basically the reverse operation of a derivative. If you have a function, its antiderivative is another function whose derivative is the original function. For example, the derivative of $x^2$ is $2x$, so an antiderivative of $2x$ is $x^2$. But here's a catch: antiderivatives aren't unique. The derivative of $x^2 + 5$ is also $2x$, so $x^2 + 5$ is another antiderivative of $2x$. In fact, there are infinitely many antiderivatives, differing only by a constant. This constant is usually represented as $C$. When we're calculating definite integrals using the FTC, this constant cancels out, so we don't need to worry about it. The notation for a definite integral looks like this: $\int_{a}^{b} f(x) dx$. The integral symbol looks like a stretched-out "S" (for "sum"), and the $a$ and $b$ are the limits of integration (the interval over which we're finding the area). The $f(x)$ is the function we're integrating, and the $dx$ indicates that we're integrating with respect to $x$. So, when you see this notation, just think "area under the curve of $f(x)$ from $x = a$ to $x = b$".
Finding the Antiderivative of f(x) = 3x + 13
Alright, now let's apply this to our specific problem. We need to find the area under the curve of $f(x) = 3x + 13$ between $x = 14$ and $x = 17$. The first step, according to the Fundamental Theorem of Calculus, is to find an antiderivative of $f(x)$. Remember, an antiderivative is a function whose derivative is $f(x)$. So, we're looking for a function $F(x)$ such that $F'(x) = 3x + 13$. To find the antiderivative, we can use the power rule for integration in reverse. The power rule for differentiation states that the derivative of $x^n$ is $nx^{n-1}$. So, to undo this, we increase the power by one and divide by the new power. Let's apply this to each term in our function. For the term $3x$, which can be written as $3x^1$, we increase the power by one to get $x^2$, and then divide by the new power (2), and don't forget the constant 3, resulting in $\frac{3}{2}x^2$. For the constant term 13, we think about what function would have a derivative of 13. Since the derivative of $kx$ is $k$, the antiderivative of 13 is simply $13x$. Putting these two pieces together, we get an antiderivative of $f(x) = 3x + 13$ as $F(x) = \frac{3}{2}x^2 + 13x$. Remember, we don't need to add the constant of integration $C$ here because it will cancel out in the next step. We've found our antiderivative! This is a crucial step, so make sure you're comfortable with the process. If you're unsure, practice finding antiderivatives of different functions. It's a skill that will come in handy throughout your calculus journey. Now that we have the antiderivative, we're ready to apply the Evaluation Theorem and find the area under the curve.
Applying the Evaluation Theorem
We've found the antiderivative, $F(x) = \frac3}{2}x^2 + 13x$. Now, it's time to use the Evaluation Theorem, which is the second part of the Fundamental Theorem of Calculus. Remember, the Evaluation Theorem tells us that the definite integral of $f(x)$ from $a$ to $b$ is simply $F(b) - F(a)$, where $F(x)$ is an antiderivative of $f(x)$. In our case, we want to find the area under the curve of $f(x) = 3x + 13$ between $x = 14$ and $x = 17$. So, $a = 14$ and $b = 17$. We need to calculate $F(17) - F(14)$. Let's start by finding $F(17)$. We substitute $x = 17$ into our antiderivative $F(x) = \frac{3}{2}x^2 + 13x$2}(17)^2 + 13(17) = \frac{3}{2}(289) + 221 = 433.5 + 221 = 654.5$. Next, we find $F(14)$. We substitute $x = 14$ into our antiderivative{2}(14)^2 + 13(14) = \frac{3}{2}(196) + 182 = 294 + 182 = 476$. Now, we subtract $F(14)$ from $F(17)$: $F(17) - F(14) = 654.5 - 476 = 178.5$. So, the area under the curve of $f(x) = 3x + 13$ between $x = 14$ and $x = 17$ is 178.5 square units. That's it! We've successfully used the Fundamental Theorem of Calculus to find the area under a curve. It might seem like a lot of steps, but once you get the hang of it, it becomes a powerful tool for solving a wide range of problems.
Conclusion
Awesome work, guys! We've tackled a challenging problem and come out on top. We used the Fundamental Theorem of Calculus to find the area under the curve of $f(x) = 3x + 13$ between $x = 14$ and $x = 17$. We started by understanding the concept of area under a curve, then we recapped the key idea of the Fundamental Theorem of Calculus, specifically the Evaluation Theorem. We found the antiderivative of our function and finally applied the Evaluation Theorem to calculate the definite integral, which gave us the area. The final answer is 178.5 square units. Remember, the beauty of calculus lies in its ability to connect abstract mathematical concepts to real-world applications. Finding the area under a curve isn't just a theoretical exercise; it's a fundamental tool in many fields. So, keep practicing, keep exploring, and keep those calculus skills sharp! You've got this!