Definite Integral Of (4x^5 + 5x^2) From -3 To 4
Let's dive into the world of calculus and explore how to solve a definite integral! In this article, we'll break down the steps to calculate the definite integral of the function f(x) = 4x^5 + 5x^2 over the interval [-3, 4]. Understanding definite integrals is crucial in various fields, including physics, engineering, and economics, as they help us determine areas, volumes, and other accumulated quantities. So, grab your thinking caps, and let's get started!
Understanding Definite Integrals
Before we jump into the calculation, let's ensure we have a solid understanding of what a definite integral represents. At its core, a definite integral calculates the signed area between a curve and the x-axis over a specified interval. The "signed" part is important because areas above the x-axis are considered positive, while areas below the x-axis are negative. This concept is fundamental in various applications, such as determining the displacement of an object given its velocity function or calculating the total revenue given a marginal revenue function.
The definite integral is denoted by the following expression:
∫[a, b] f(x) dx
Where:
- ∫ represents the integral symbol.
- a and b are the limits of integration, defining the interval over which we're calculating the area.
- f(x) is the function we're integrating (the integrand).
- dx indicates that we're integrating with respect to the variable x.
The Fundamental Theorem of Calculus provides the key to evaluating definite integrals. It states that if F(x) is an antiderivative of f(x) (meaning F'(x) = f(x)), then:
∫[a, b] f(x) dx = F(b) - F(a)
In simpler terms, to evaluate a definite integral, we first find the antiderivative of the function, then evaluate the antiderivative at the upper and lower limits of integration, and finally subtract the value at the lower limit from the value at the upper limit. This theorem connects the concepts of differentiation and integration, making it a cornerstone of calculus.
Step-by-Step Calculation
Now that we've refreshed our understanding of definite integrals, let's tackle the specific problem at hand: finding the definite integral of f(x) = 4x^5 + 5x^2 from -3 to 4.
Step 1: Find the Antiderivative
The first step is to determine the antiderivative, F(x), of the function f(x) = 4x^5 + 5x^2. Recall that an antiderivative is a function whose derivative is equal to the original function. To find the antiderivative, we'll use the power rule of integration, which states:
∫x^n dx = (x^(n+1))/(n+1) + C, where n ≠-1 and C is the constant of integration.
Applying the power rule to each term in f(x):
∫4x^5 dx = 4 * ∫x^5 dx = 4 * (x^6 / 6) = (2/3)x^6
∫5x^2 dx = 5 * ∫x^2 dx = 5 * (x^3 / 3) = (5/3)x^3
Therefore, the antiderivative F(x) is:
F(x) = (2/3)x^6 + (5/3)x^3 + C
Notice the constant of integration, C. While it's essential to include it when finding indefinite integrals, it cancels out when evaluating definite integrals, so we can omit it in this case. We have successfully found the antiderivative of the function, setting the stage for the next step in calculating the definite integral.
Step 2: Evaluate the Antiderivative at the Limits of Integration
Now that we have the antiderivative, F(x) = (2/3)x^6 + (5/3)x^3, the next step is to evaluate it at the limits of integration, which are -3 and 4. This means we need to calculate F(4) and F(-3).
Let's start with F(4):
F(4) = (2/3)(4^6) + (5/3)(4^3) = (2/3)(4096) + (5/3)(64) = 8192/3 + 320/3 = 8512/3
Now, let's calculate F(-3):
F(-3) = (2/3)((-3)^6) + (5/3)((-3)^3) = (2/3)(729) + (5/3)(-27) = 1458/3 - 135/3 = 1323/3 = 441
We've successfully evaluated the antiderivative at both the upper and lower limits of integration. These values will be crucial in the final step of calculating the definite integral, where we'll find the difference between F(4) and F(-3). The meticulous calculation in this step ensures the accuracy of our final result. This careful evaluation lays the groundwork for the final calculation, bringing us closer to the solution.
Step 3: Calculate the Definite Integral
The final step in calculating the definite integral is to subtract the value of the antiderivative at the lower limit of integration from its value at the upper limit. In other words, we need to calculate F(4) - F(-3). We've already found these values in the previous step:
F(4) = 8512/3
F(-3) = 441
So, the definite integral is:
∫[-3, 4] (4x^5 + 5x^2) dx = F(4) - F(-3) = 8512/3 - 441
To subtract these values, we need a common denominator. We can rewrite 441 as a fraction with a denominator of 3:
441 = 441/1 = (441 * 3) / (1 * 3) = 1323/3
Now we can perform the subtraction:
8512/3 - 1323/3 = (8512 - 1323) / 3 = 7189/3
Therefore, the definite integral of (4x^5 + 5x^2) from -3 to 4 is 7189/3. This value represents the signed area between the curve of the function and the x-axis over the interval [-3, 4]. We have successfully navigated the steps of finding the antiderivative, evaluating it at the limits of integration, and calculating the final result. This demonstrates the power of calculus in solving real-world problems involving accumulated quantities.
Conclusion
In this article, we've walked through the process of calculating the definite integral of f(x) = 4x^5 + 5x^2 from -3 to 4. We started by understanding the concept of definite integrals and the Fundamental Theorem of Calculus. Then, we systematically found the antiderivative, evaluated it at the limits of integration, and calculated the final result, which is 7189/3. This exercise showcases the practical application of calculus in determining areas and accumulated quantities. Understanding these concepts is essential for various disciplines, making the mastery of definite integrals a valuable skill.
To further enhance your understanding of definite integrals and explore more examples, consider visiting resources like Khan Academy's Calculus section.
