Calculate Definite Integral Of [f(x) + 2x + 3] With Regions Of Area 3

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In this article, we delve into the fascinating realm of calculus, specifically focusing on definite integrals and their relationship to areas bounded by curves. We will explore a problem where the graph of a function f intersects the x-axis, creating distinct regions with equal areas. Our objective is to determine the value of the definite integral of a composite function involving f, a linear term (2x), and a constant (3). This exploration will not only reinforce your understanding of integral calculus but also highlight its practical applications in geometry and problem-solving.

Problem Statement

Imagine a continuous function f whose graph intersects the x-axis at multiple points, forming three distinct regions labeled A, B, and C. Each of these regions is bounded by the graph of f and the x-axis and possesses an equal area of 3 square units. Given this information, our task is to find the value of the definite integral of the expression [ f(x) + 2x + 3 ] over a specified interval. To tackle this problem effectively, we'll need to leverage our knowledge of definite integrals, areas under curves, and the properties of integration.

Understanding the Fundamentals

Before diving into the solution, let's solidify our understanding of the fundamental concepts involved.

Definite Integrals and Areas

The definite integral of a function f(x) over an interval [a, b], denoted as ∫[a, b] f(x) dx, represents the signed area between the graph of f(x) and the x-axis. The term "signed area" is crucial here. Areas above the x-axis are considered positive, while areas below the x-axis are considered negative. This convention is essential for accurately calculating the net area and, consequently, the definite integral.

For instance, if the graph of f(x) lies entirely above the x-axis within the interval [a, b], the definite integral ∫[a, b] f(x) dx will be a positive value equal to the area enclosed between the curve and the x-axis. Conversely, if the graph lies entirely below the x-axis, the definite integral will be a negative value, the magnitude of which corresponds to the area enclosed. When the graph crosses the x-axis within the interval, the definite integral represents the algebraic sum of the areas above and below the axis.

Properties of Definite Integrals

Definite integrals possess several key properties that facilitate their manipulation and evaluation. These properties are invaluable tools for simplifying complex integrals and breaking them down into manageable parts. Some of the most important properties include:

  1. Linearity: The integral of a sum (or difference) of functions is equal to the sum (or difference) of their individual integrals. Mathematically, this can be expressed as:

    ∫[a, b] [ f(x) + g(x) ] dx = ∫[a, b] f(x) dx + ∫[a, b] g(x) dx

    This property allows us to split complex integrands into simpler components, making the integration process more straightforward.

  2. Constant Multiple: The integral of a constant multiplied by a function is equal to the constant multiplied by the integral of the function. Expressed mathematically:

    ∫[a, b] c f(x) dx = c ∫[a, b] f(x) dx, where c is a constant.

    This property enables us to extract constant factors from the integral, simplifying the calculation.

  3. Additivity over Intervals: If c is a point within the interval [a, b], then the integral over [a, b] can be split into the sum of integrals over [a, c] and [c, b]:

    ∫[a, b] f(x) dx = ∫[a, c] f(x) dx + ∫[c, b] f(x) dx

    This property is particularly useful when dealing with piecewise functions or when the integrand exhibits different behaviors over different subintervals.

  4. Integral of x: The definite integral of the linear function x over the interval [a, b] can be calculated directly using the power rule of integration:

    ∫[a, b] x dx = (b^2 / 2) - (a^2 / 2)

    This result is a fundamental building block for integrating more complex expressions involving x.

  5. Integral of a Constant: The definite integral of a constant k over the interval [a, b] is simply the product of the constant and the length of the interval:

    ∫[a, b] k dx = k (b - a)

    This is a straightforward but essential property that simplifies the integration of constant terms.

Applying the Properties to the Problem

Now that we've reviewed the fundamental properties of definite integrals, let's apply them to our specific problem. We are tasked with finding the value of:

∫[a, b] [ f(x) + 2x + 3 ] dx

where the interval [a, b] encompasses the regions A, B, and C. Using the linearity property of integrals, we can split this integral into three separate integrals:

∫[a, b] [ f(x) + 2x + 3 ] dx = ∫[a, b] f(x) dx + ∫[a, b] 2x dx + ∫[a, b] 3 dx

This separation allows us to address each component individually.

Analyzing the Regions and Areas

The problem states that the regions A, B, and C, bounded by the graph of f and the x-axis, each have an area of 3. However, it's crucial to remember the concept of signed areas. Since some regions may lie below the x-axis, their corresponding definite integrals will be negative.

Let's assume the regions are arranged such that region A is above the x-axis, region B is below, and region C is above. This is a common scenario in such problems. Consequently, we have:

  • ∫[A] f(x) dx = 3 (positive, since region A is above the x-axis)
  • ∫[B] f(x) dx = -3 (negative, since region B is below the x-axis)
  • ∫[C] f(x) dx = 3 (positive, since region C is above the x-axis)

To find the integral of f(x) over the entire interval [a, b], which spans all three regions, we sum the integrals over each region:

∫[a, b] f(x) dx = ∫[A] f(x) dx + ∫[B] f(x) dx + ∫[C] f(x) dx = 3 + (-3) + 3 = 3

Thus, the definite integral of f(x) over the entire interval [a, b] is 3.

Evaluating the Remaining Integrals

Now that we've determined the value of ∫[a, b] f(x) dx, we need to evaluate the remaining integrals in our expression:

∫[a, b] 2x dx and ∫[a, b] 3 dx

Evaluating ∫[a, b] 2x dx

Using the constant multiple property, we can rewrite this integral as:

∫[a, b] 2x dx = 2 ∫[a, b] x dx

We know that the integral of x over the interval [a, b] is (b^2 / 2) - (a^2 / 2). Therefore,

2 ∫[a, b] x dx = 2 [ (b^2 / 2) - (a^2 / 2) ] = b^2 - a^2

Evaluating ∫[a, b] 3 dx

This is the integral of a constant. Using the property of the integral of a constant, we have:

∫[a, b] 3 dx = 3 (b - a)

Putting It All Together

We have now evaluated all the individual integrals. Let's substitute these values back into our original expression:

∫[a, b] [ f(x) + 2x + 3 ] dx = ∫[a, b] f(x) dx + ∫[a, b] 2x dx + ∫[a, b] 3 dx

= 3 + (b^2 - a^2) + 3(b - a)

Therefore, the value of the definite integral ∫[a, b] [ f(x) + 2x + 3 ] dx is 3 + (b^2 - a^2) + 3(b - a).

Conclusion

In this article, we successfully determined the value of the definite integral of [ f(x) + 2x + 3 ] given information about the areas bounded by the graph of f and the x-axis. We achieved this by applying the fundamental properties of definite integrals, understanding the concept of signed areas, and breaking down the problem into manageable steps. This exercise highlights the power of integral calculus in solving geometric problems and underscores the importance of a solid understanding of its underlying principles.

This problem serves as a valuable illustration of how definite integrals can be used to calculate areas and how the properties of integrals can simplify complex calculations. By mastering these concepts, you'll be well-equipped to tackle a wide range of calculus problems and appreciate the elegance and power of this mathematical tool.