Evaluate The Definite Integral Of (10x + 6) From 4 To 9 A Step-by-Step Guide

Evaluating definite integrals might seem daunting at first, but trust me, guys, it's a super useful skill in calculus and has tons of applications in physics, engineering, and even economics. In this article, we're going to break down how to evaluate the definite integral of a simple polynomial function, specifically $\int_4^9(10x + 6) dx$. We'll go through each step slowly and methodically, so you'll not only get the answer but also understand the why behind the how. So, let's dive in and conquer this integral together!

Understanding Definite Integrals

Before we jump into the calculation, let's quickly recap what a definite integral actually represents. Think of it as finding the area under a curve between two specific points on the x-axis. These points are our limits of integration – in this case, 4 and 9. The function we're integrating, $10x + 6$, defines the curve. So, the definite integral $\int_4^9(10x + 6) dx$ gives us the area bounded by the line $y = 10x + 6$, the x-axis, and the vertical lines $x = 4$ and $x = 9$.

The definite integral is a fundamental concept in calculus, representing the accumulated change of a function over an interval. It has wide-ranging applications, including calculating areas, volumes, work done by a force, and average values. Understanding the mechanics of evaluating these integrals is thus crucial for various scientific and engineering disciplines. The integral we're tackling, $\int_4^9(10x + 6) dx$, is a classic example involving a linear function, making it an ideal starting point for grasping the concept. The limits of integration, 4 and 9, define the specific interval over which we are calculating this accumulated change. Geometrically, this integral represents the area under the line $y = 10x + 6$ bounded by the x-axis and the vertical lines $x = 4$ and $x = 9$. This area can be visualized as a trapezoid, and indeed, the result of the integration will correspond to the area of this trapezoid. However, the power of integration lies in its ability to handle more complex functions and shapes where simple geometric formulas are insufficient. Therefore, mastering the techniques of integration is essential for tackling a wide array of problems in mathematics and its applications.

Step 1: Find the Indefinite Integral

The first step in evaluating a definite integral is to find the indefinite integral of the function. The indefinite integral is essentially the antiderivative – a function whose derivative is the function we're integrating. To find the indefinite integral of $10x + 6$, we'll use the power rule and the constant multiple rule of integration. Remember the power rule? It states that $\int x^n dx = \frac{x^{n+1}}{n+1} + C$, where C is the constant of integration. And the constant multiple rule says that $\int kf(x) dx = k \int f(x) dx$, where k is a constant.

So, let's break it down:

• $\int 10x dx = 10 \int x^1 dx = 10 * \frac{x^{1+1}}{1+1} = 10 * \frac{x^2}{2} = 5x^2$
• $\int 6 dx = 6 \int x^0 dx = 6 * \frac{x^{0+1}}{0+1} = 6x$

Combining these, the indefinite integral of $10x + 6$ is $5x^2 + 6x + C$. Don't forget that “+ C”! It's crucial because the derivative of any constant is zero, so there could be infinitely many constants that satisfy the antiderivative. The indefinite integral is the foundation upon which we build the definite integral. It represents the family of functions whose derivative is the original function we are integrating. For $10x + 6$, we use the power rule of integration, which is a cornerstone technique. This rule allows us to reverse the process of differentiation, effectively finding the function that, when differentiated, yields $10x + 6$. We also apply the constant multiple rule, which allows us to pull constant coefficients out of the integral, simplifying the calculation. The result, $5x^2 + 6x + C$, encapsulates all possible antiderivatives of $10x + 6$, differing only by a constant. This constant, denoted by 'C', is a critical part of the indefinite integral because it accounts for the fact that the derivative of a constant is always zero. Therefore, when we reverse the process of differentiation, we must acknowledge the potential presence of this arbitrary constant. Understanding the indefinite integral is paramount, as it sets the stage for the next step: evaluating the definite integral using the Fundamental Theorem of Calculus. This theorem provides the precise link between indefinite and definite integrals, allowing us to compute the exact area under a curve.

Step 2: Apply the Fundamental Theorem of Calculus

Now comes the magic step where we use the Fundamental Theorem of Calculus (FTC). This theorem provides a neat way to evaluate definite integrals. It states that if F(x) is the indefinite integral of f(x), then $\int_a^b f(x) dx = F(b) - F(a)$. In plain English, it means we plug in the upper limit (b) into the indefinite integral, plug in the lower limit (a) into the indefinite integral, and then subtract the second result from the first. Super straightforward, right?

In our case, $f(x) = 10x + 6$, the indefinite integral $F(x) = 5x^2 + 6x + C$, $a = 4$, and $b = 9$. Let's plug those values into the FTC formula:

$\int_4^9 (10x + 6) dx = F(9) - F(4) = (5(9)^2 + 6(9) + C) - (5(4)^2 + 6(4) + C)$ The Fundamental Theorem of Calculus is the linchpin that connects differentiation and integration, allowing us to precisely compute definite integrals. This theorem elegantly states that the definite integral of a function $f(x)$ from $a$ to $b$ is simply the difference in the values of its antiderivative $F(x)$ evaluated at the limits of integration, i.e., $F(b) - F(a)$. This seemingly simple formula is a powerful tool, transforming the problem of finding the area under a curve into a straightforward evaluation of a function at two points. In our specific case, we have already found the indefinite integral $F(x) = 5x^2 + 6x + C$. Now, we apply the FTC by substituting the upper limit of integration, 9, and the lower limit, 4, into $F(x)$. This yields two expressions: $F(9) = 5(9)^2 + 6(9) + C$ and $F(4) = 5(4)^2 + 6(4) + C$. The constant of integration, 'C', appears in both expressions, which is a crucial observation. As we proceed to subtract $F(4)$ from $F(9)$, this constant will cancel out, demonstrating a key property of definite integrals – they are independent of the constant of integration. This simplification makes the calculation much cleaner and highlights the elegance of the Fundamental Theorem of Calculus.

Step 3: Evaluate and Simplify

Time to do the math! Let's simplify that expression we got from the FTC:

$(5(9)^2 + 6(9) + C) - (5(4)^2 + 6(4) + C) = (5(81) + 54 + C) - (5(16) + 24 + C)$

$= (405 + 54 + C) - (80 + 24 + C) = 459 + C - 104 - C$

Notice how the 'C' terms cancel out? This is why we don't actually need to worry about the constant of integration when evaluating definite integrals. Cool, huh?

Continuing the simplification: $459 - 104 = 355$

So, the value of the definite integral $\int_4^9(10x + 6) dx$ is 355.

This step involves the arithmetic heart of the evaluation, where we meticulously compute the values obtained from applying the Fundamental Theorem of Calculus. We begin by expanding the squares and products within the expressions $F(9)$ and $F(4)$. This yields $(5(81) + 54 + C) - (5(16) + 24 + C)$, which further simplifies to $(405 + 54 + C) - (80 + 24 + C)$. Now comes a crucial observation: the constant of integration, 'C', which we dutifully carried along, is about to vanish. When we distribute the subtraction, we get $405 + 54 + C - 80 - 24 - C$. The '+C' and '-C' terms neatly cancel each other out. This cancellation is not a coincidence; it's a fundamental property of definite integrals. Because we are finding the difference between the antiderivative evaluated at two points, any constant term will always cancel out. This means that for definite integrals, we can often ignore the constant of integration during the initial antiderivative step, streamlining our calculations. With the 'C' terms gone, we are left with a simple arithmetic problem: $405 + 54 - 80 - 24$. Performing these operations in order, we first add 405 and 54 to get 459, then subtract 80 to get 379, and finally subtract 24 to arrive at the final answer: 355. This numerical result represents the exact area under the curve $y = 10x + 6$ between the lines $x = 4$ and $x = 9$.

Conclusion

There you have it! We've successfully evaluated the definite integral $\int_4^9(10x + 6) dx$, and the answer is 355. We tackled this by first finding the indefinite integral, then applying the Fundamental Theorem of Calculus, and finally, simplifying the expression. Remember, the key to mastering definite integrals is practice. So, try out some more examples, and you'll be a pro in no time!

Evaluating definite integrals is a core skill in calculus, and as we've demonstrated, it's a process that can be broken down into manageable steps. We began by understanding the conceptual meaning of a definite integral, which is essentially the area under a curve between two specified limits. This geometric interpretation provides a visual intuition for what we are calculating. Then, we moved on to the mechanics of the calculation. The first step involves finding the indefinite integral, which is the antiderivative of the function. This step requires a solid understanding of integration rules, such as the power rule and the constant multiple rule. The constant of integration, 'C', is a crucial part of the indefinite integral, although it ultimately cancels out in the definite integral calculation. Next, we applied the Fundamental Theorem of Calculus, the cornerstone of our evaluation. This theorem allows us to compute the definite integral by simply finding the difference in the values of the antiderivative at the upper and lower limits of integration. This step transforms the problem from finding an area to a straightforward function evaluation. Finally, we simplified the resulting expression, performing the necessary arithmetic to arrive at the numerical answer. In our example, the definite integral $\int_4^9(10x + 6) dx$ evaluates to 355. This result represents the precise area under the line $y = 10x + 6$ between $x = 4$ and $x = 9$. Mastering this process empowers you to tackle a wide range of problems involving accumulated change and areas, making it an indispensable tool in various scientific and engineering fields.