Area Between Curves Calculation Of Region Bounded By F(x) And G(x)

Introduction to Calculating Area Between Curves

In calculus, determining the area of a region bounded by curves is a fundamental concept with wide-ranging applications. In this article, we will focus on finding the precise area of a region R delimited by two specific functions: f(x) = 2sin(x) + 2 and g(x) = (4x/π) + 2. We will carry out this calculation over the interval [0, π/2]. This type of problem is quintessential in integral calculus, as it elegantly combines trigonometric and linear functions, requiring a solid grasp of both integration techniques and geometrical interpretation. To accurately compute the area, we will employ definite integrals, a powerful tool for summing infinitesimally small areas to arrive at the total area enclosed between the curves. Understanding how to set up and solve these integrals is crucial not only for academic purposes but also for practical applications in engineering, physics, and economics, where calculating areas and volumes is paramount. So, let's delve into the step-by-step process of finding this area, ensuring we arrive at an exact solution, which is a hallmark of mathematical rigor and precision.

Problem Statement: Defining the Region R

Before we dive into the mathematical computations, let's clearly define the region R we're interested in. The region is bounded by two functions: f(x) = 2sin(x) + 2 and g(x) = (4x/π) + 2. The function f(x) is a sinusoidal function, specifically a sine wave that has been vertically stretched by a factor of 2 and then shifted upwards by 2 units. This vertical shift ensures that the sine wave oscillates around the line y = 2 rather than the x-axis. On the other hand, the function g(x) represents a linear equation, a straight line with a slope of 4/π and a y-intercept of 2. This line intersects the y-axis at the point (0, 2) and slopes upwards as x increases.

The interval of interest is [0, π/2], which means we are only considering the region bounded by these two functions within this specific range of x values. This interval is crucial because it defines the limits of integration, telling us where to start and stop summing the area between the curves. Geometrically, this interval represents a portion of the first quadrant of the Cartesian plane. Understanding these boundaries is key to setting up the definite integral correctly. We need to identify which function is above the other within this interval, as this will determine the order of subtraction in our integral. A clear visualization of these functions within the specified interval helps in accurately setting up the integral and avoiding common pitfalls in area calculation. This careful setup ensures that we accurately capture the area enclosed between the curves, leading to a precise final answer.

Setting up the Definite Integral: A Step-by-Step Guide

To calculate the area of the region R bounded by the functions f(x) = 2sin(x) + 2 and g(x) = (4x/π) + 2 over the interval [0, π/2], we need to set up a definite integral. The fundamental principle behind this approach is to integrate the difference between the two functions over the given interval. This difference represents the height of a thin rectangle at each point x, and the integral sums up the areas of all such rectangles to give the total area between the curves.

The first crucial step is to determine which function is greater than the other within the interval [0, π/2]. This is important because we need to subtract the lower function from the higher function to ensure the area is positive. By analyzing the functions or sketching their graphs, we can observe that f(x) = 2sin(x) + 2 is above g(x) = (4x/π) + 2 in this interval. This means we will subtract g(x) from f(x).

Now we can set up the definite integral. The area A of the region R is given by:

A = ∫[0 to π/2] (f(x) - g(x)) dx

Substituting the functions f(x) and g(x), we get:

A = ∫[0 to π/2] ((2sin(x) + 2) - (4x/π + 2)) dx

Simplifying the integrand, we have:

A = ∫[0 to π/2] (2sin(x) - 4x/π) dx

This integral represents the exact area we want to find. The next step is to evaluate this integral, which involves finding the antiderivatives of the terms inside the integral and then applying the limits of integration. This setup is the foundation for the rest of the calculation, so ensuring it is correct is paramount. With the integral set up correctly, we are now ready to proceed with the evaluation, which will lead us to the final exact answer for the area.

Evaluating the Definite Integral: Finding the Exact Area

With the definite integral set up as A = ∫[0 to π/2] (2sin(x) - 4x/π) dx, our next step is to evaluate it to find the exact area of the region R. This involves finding the antiderivative of the integrand and then applying the limits of integration.

First, let's find the antiderivative of each term in the integrand. The antiderivative of 2sin(x) is -2cos(x), and the antiderivative of -4x/π is -(2x^2)/π. So, the antiderivative of the entire integrand is:

-2cos(x) - (2x^2)/π

Now, we apply the limits of integration, which are 0 and π/2. This means we will evaluate the antiderivative at π/2 and then subtract its value at 0.

Evaluating at x = π/2, we get:

-2cos(π/2) - (2(π/2)^2)/π = -2(0) - (2(π^2/4))/π = 0 - π/2 = -π/2

Evaluating at x = 0, we get:

-2cos(0) - (2(0)^2)/π = -2(1) - 0 = -2

Now, we subtract the value at the lower limit from the value at the upper limit:

A = (-π/2) - (-2) = -π/2 + 2

So, the exact area of the region R is 2 - π/2. This is a precise value, expressed in terms of π, which is essential for maintaining mathematical accuracy. This result represents the culmination of our calculus problem, providing a definitive answer to the area bounded by the given functions over the specified interval. The final step is to express this answer with the correct units, which we will do in the concluding section.

Expressing the Final Answer: Units and Conclusion

Having computed the exact area of the region R as 2 - π/2, the final step is to express this answer with the appropriate units and provide a concluding statement. In mathematics, especially in applied contexts, specifying the units is crucial for the answer to be complete and meaningful.

Since we are calculating an area, the units will be in square units. If the original functions were defined in terms of, say, meters, then the area would be in square meters. However, without specific units provided in the problem statement, we express the area in general terms as "units squared" or "square units".

Therefore, the exact area of the region R bounded by the functions f(x) = 2sin(x) + 2 and g(x) = (4x/π) + 2 on the interval [0, π/2] is:

A = 2 - π/2 square units

This result is an exact value, as it is expressed in terms of π and not a decimal approximation. Exact answers are preferred in mathematics because they preserve precision and avoid rounding errors. In conclusion, we have successfully found the area of the region R by setting up and evaluating a definite integral. This process involved understanding the geometry of the functions, determining the correct order of subtraction, finding antiderivatives, and applying the limits of integration. The final answer, 2 - π/2 square units, represents the precise area enclosed between the two curves within the given interval, showcasing the power and elegance of integral calculus in solving real-world problems.