Estimating Area Under Curve Y=3+2x-x^2 With Trapezoidal Rule

Jul 10, 2025 by ADMIN 61 views

Estimate the Area Under the Curve y=3+2x-x^2 using the Trapezoidal Rule

Estimating the area under a curve is a fundamental problem in calculus, with applications ranging from physics to economics. In this article, we will explore how to approximate the area under the curve y = 3 + 2x - x² over the interval [-1, 2] using the trapezoidal rule. The trapezoidal rule is a numerical integration technique that approximates the definite integral of a function by dividing the area under the curve into trapezoids and summing their areas. This method is particularly useful when finding an exact solution through analytical methods is difficult or impossible. We will use 3 trapezoids of equal height to achieve a balance between accuracy and computational effort. Let's delve into the steps involved in applying the trapezoidal rule and discuss the underlying concepts.

Understanding the Trapezoidal Rule

The trapezoidal rule approximates the area under a curve by dividing the region into trapezoids and summing their areas. This method is a numerical integration technique, which means it provides an approximate solution to a definite integral. The core idea behind the trapezoidal rule is to replace the curve of the function with straight line segments, forming trapezoids. The area of each trapezoid is then calculated using the formula for the area of a trapezoid, and these areas are summed to estimate the total area under the curve. This approach is particularly useful when finding an exact solution through analytical methods is challenging or impossible.

To effectively apply the trapezoidal rule, we need to understand the key parameters involved. First, we have the interval of integration, which defines the boundaries over which we want to calculate the area under the curve. This interval is denoted as [a, b], where a is the lower limit and b is the upper limit. Next, we need to determine the number of trapezoids, n, we will use to approximate the area. A larger number of trapezoids generally leads to a more accurate approximation, as the straight line segments better fit the curve. However, increasing the number of trapezoids also increases the computational effort required. Once we have the interval and the number of trapezoids, we can calculate the width of each trapezoid, Δx, using the formula: Δx = (b - a) / n. This width represents the base of each trapezoid and is crucial for calculating the area. Finally, we need to evaluate the function at the endpoints of each subinterval to determine the heights of the trapezoids. These heights, along with the width, allow us to calculate the area of each trapezoid and ultimately approximate the total area under the curve.

The formula for the trapezoidal rule is derived from the area formula of a trapezoid, which is given by Area = (1/2) * (base) * (height1 + height2). In the context of the trapezoidal rule, the base of each trapezoid is the width Δx, and the heights are the function values at the endpoints of the subintervals. By summing the areas of all the trapezoids, we arrive at the general formula for the trapezoidal rule:

∫ab f(x) dx ≈ (Δx / 2) * [f(x0) + 2f(x1) + 2f(x2) + ... + 2f(xn-1) + f(xn)]

Where:

a is the lower limit of integration.
b is the upper limit of integration.
n is the number of trapezoids.
Δx = (b - a) / n is the width of each trapezoid.
x0, x1, x2, ..., xn are the endpoints of the subintervals, with x0 = a and xn = b.
f(xi) represents the function value at xi.

This formula essentially averages the function values at the endpoints of each subinterval and multiplies the result by the width of the trapezoids. The factor of 2 in front of the interior function values accounts for the fact that these values are used as heights for two adjacent trapezoids. The trapezoidal rule provides a practical and widely used method for approximating definite integrals, particularly when analytical solutions are not readily available.

Applying the Trapezoidal Rule to y=3+2x-x²

To estimate the area under the curve y = 3 + 2x - x² over the interval [-1, 2] using the trapezoidal rule with 3 trapezoids, we need to follow a step-by-step process. This involves determining the width of each trapezoid, identifying the x-coordinates at which to evaluate the function, calculating the corresponding y-values, and finally, applying the trapezoidal rule formula. By carefully executing these steps, we can obtain a reliable approximation of the area under the curve.

First, let's calculate the width of each trapezoid. We are given the interval [-1, 2], so a = -1 and b = 2. We are also given that we should use 3 trapezoids, so n = 3. The width Δx can be calculated using the formula:

Δx = (b - a) / n = (2 - (-1)) / 3 = 3 / 3 = 1

This means each trapezoid will have a width of 1 unit along the x-axis. Next, we need to determine the x-coordinates at which to evaluate the function. These x-coordinates will be the vertices of our trapezoids along the x-axis. Starting from a = -1, we add Δx = 1 successively to find the x-coordinates:

x0 = -1
x1 = -1 + 1 = 0
x2 = 0 + 1 = 1
x3 = 1 + 1 = 2

Thus, our x-coordinates are -1, 0, 1, and 2. Now, we need to calculate the corresponding y-values by evaluating the function y = 3 + 2x - x² at these x-coordinates:

f(x0) = f(-1) = 3 + 2(-1) - (-1)² = 3 - 2 - 1 = 0
f(x1) = f(0) = 3 + 2(0) - (0)² = 3 + 0 - 0 = 3
f(x2) = f(1) = 3 + 2(1) - (1)² = 3 + 2 - 1 = 4
f(x3) = f(2) = 3 + 2(2) - (2)² = 3 + 4 - 4 = 3

So, the y-values corresponding to our x-coordinates are 0, 3, 4, and 3. With all the necessary components calculated, we can now apply the trapezoidal rule formula to estimate the area under the curve. The trapezoidal rule formula is:

∫ab f(x) dx ≈ (Δx / 2) * [f(x0) + 2f(x1) + 2f(x2) + f(x3)]

Plugging in the values we calculated:

Area ≈ (1 / 2) * [0 + 2(3) + 2(4) + 3] Area ≈ (1 / 2) * [0 + 6 + 8 + 3] Area ≈ (1 / 2) * [17] Area ≈ 17 / 2

Therefore, using the trapezoidal rule with 3 trapezoids, we estimate the area under the curve y = 3 + 2x - x² over the interval [-1, 2] to be 17/2. This step-by-step application demonstrates the practical use of the trapezoidal rule in approximating definite integrals.

Step-by-Step Calculation

To solidify our understanding of the trapezoidal rule and its application, let's break down the calculation process into clear, manageable steps. This methodical approach will ensure accuracy and clarity in our estimation of the area under the curve y = 3 + 2x - x² over the interval [-1, 2] using 3 trapezoids. Each step is crucial and builds upon the previous one, ultimately leading to our final approximation.

The first step in applying the trapezoidal rule is to determine the width of each trapezoid. This width, denoted as Δx, is essential for calculating the area of each individual trapezoid and, consequently, the total estimated area. We use the formula Δx = (b - a) / n, where a and b are the lower and upper limits of the interval, respectively, and n is the number of trapezoids. In our case, a = -1, b = 2, and n = 3. Plugging these values into the formula, we get:

Δx = (2 - (-1)) / 3 = 3 / 3 = 1

Thus, the width of each trapezoid is 1 unit. This means that each trapezoid will span 1 unit along the x-axis, dividing the interval [-1, 2] into three equal subintervals. Knowing the width is crucial for the next steps, as it forms the base of each trapezoid in our approximation.

The next critical step is to identify the x-coordinates at which we need to evaluate the function. These x-coordinates will serve as the vertices of our trapezoids along the x-axis and are the points at which we will calculate the height of each trapezoid. We start with the lower limit of the interval, a = -1, and successively add the width Δx = 1 to find the remaining x-coordinates. This process ensures that we cover the entire interval with our trapezoids. The x-coordinates are calculated as follows:

x0 = a = -1
x1 = x0 + Δx = -1 + 1 = 0
x2 = x1 + Δx = 0 + 1 = 1
x3 = x2 + Δx = 1 + 1 = 2

Therefore, our x-coordinates are -1, 0, 1, and 2. These values represent the points along the x-axis where we will evaluate the function to determine the corresponding y-values, which will serve as the heights of our trapezoids. Accurate identification of these x-coordinates is essential for the correct application of the trapezoidal rule.

Now that we have the x-coordinates, we need to calculate the corresponding y-values by evaluating the function y = 3 + 2x - x² at each of these points. These y-values represent the heights of the trapezoids at the specified x-coordinates and are crucial for calculating the area of each trapezoid. We substitute each x-coordinate into the function and calculate the resulting y-value:

f(x0) = f(-1) = 3 + 2(-1) - (-1)² = 3 - 2 - 1 = 0
f(x1) = f(0) = 3 + 2(0) - (0)² = 3 + 0 - 0 = 3
f(x2) = f(1) = 3 + 2(1) - (1)² = 3 + 2 - 1 = 4
f(x3) = f(2) = 3 + 2(2) - (2)² = 3 + 4 - 4 = 3

Thus, the y-values corresponding to our x-coordinates are 0, 3, 4, and 3. These values represent the heights of the trapezoids at x = -1, 0, 1, and 2, respectively. With these heights and the width of the trapezoids, we are now ready to apply the trapezoidal rule formula to estimate the area under the curve.

The final step is to apply the trapezoidal rule formula to estimate the area under the curve. This formula combines the width of the trapezoids and the function values at the x-coordinates to provide an approximation of the definite integral. The trapezoidal rule formula is given by:

∫ab f(x) dx ≈ (Δx / 2) * [f(x0) + 2f(x1) + 2f(x2) + f(x3)]

We have already calculated all the necessary components: Δx = 1, f(x0) = 0, f(x1) = 3, f(x2) = 4, and f(x3) = 3. Plugging these values into the formula, we get:

Area ≈ (1 / 2) * [0 + 2(3) + 2(4) + 3] Area ≈ (1 / 2) * [0 + 6 + 8 + 3] Area ≈ (1 / 2) * [17] Area ≈ 17 / 2

Therefore, using the trapezoidal rule with 3 trapezoids, we estimate the area under the curve y = 3 + 2x - x² over the interval [-1, 2] to be 17/2. This step-by-step calculation demonstrates the practical application of the trapezoidal rule, breaking down the process into manageable parts to ensure accuracy and understanding.

Analyzing the Result

After applying the trapezoidal rule, we obtained an estimated area of 17/2 under the curve y = 3 + 2x - x² over the interval [-1, 2]. This result provides a numerical approximation of the definite integral of the function over the given interval. However, it's important to understand that this is an approximation, and the accuracy of the trapezoidal rule depends on several factors, including the number of trapezoids used and the nature of the function itself. Analyzing the result involves considering the implications of the approximation and comparing it with other possible methods or known results.

One key aspect of analyzing the result is to consider the potential for error. The trapezoidal rule approximates the area under the curve by using straight line segments to form trapezoids. This means that if the curve is not a straight line, there will be some discrepancy between the actual area and the approximated area. The error arises from the fact that the trapezoids may either overestimate or underestimate the area, depending on the concavity of the curve. In general, increasing the number of trapezoids reduces the error, as the straight line segments more closely follow the curve. However, there is always some inherent error associated with numerical integration techniques like the trapezoidal rule.

In our case, we used 3 trapezoids to approximate the area. While this provides a reasonable estimate, using more trapezoids would likely result in a more accurate approximation. The error in the trapezoidal rule is related to the second derivative of the function. If the second derivative is large, the curve is more curved, and the trapezoidal rule may have a larger error. Conversely, if the second derivative is small, the curve is closer to a straight line, and the trapezoidal rule will provide a more accurate approximation. To assess the error in our approximation, we could compare our result with the exact value obtained through analytical integration or by using a higher number of trapezoids.

Another important aspect of analyzing the result is to compare it with other approximation methods. Besides the trapezoidal rule, other numerical integration techniques, such as the midpoint rule and Simpson's rule, can be used to estimate the area under a curve. Each method has its own strengths and weaknesses, and the choice of method may depend on the specific function and the desired level of accuracy. For example, Simpson's rule generally provides a more accurate approximation than the trapezoidal rule for the same number of subintervals, as it uses quadratic approximations instead of linear approximations. Comparing the results obtained from different methods can provide valuable insights into the accuracy and reliability of our approximation.

Finally, it's beneficial to interpret the result in the context of the problem. The area under the curve y = 3 + 2x - x² over the interval [-1, 2] represents the definite integral of the function over that interval. This definite integral has a specific meaning depending on the application. For example, if y represents the rate of flow of a fluid, the area under the curve would represent the total amount of fluid that has flowed over the interval [-1, 2]. Similarly, if y represents the velocity of an object, the area under the curve would represent the displacement of the object over the interval. Understanding the context of the problem allows us to interpret the numerical result in a meaningful way and to assess its significance.

In summary, analyzing the result obtained from the trapezoidal rule involves considering the potential for error, comparing it with other approximation methods, and interpreting it in the context of the problem. By carefully examining these aspects, we can gain a deeper understanding of the approximation and its implications.

Conclusion

In this article, we have successfully estimated the area under the curve y = 3 + 2x - x² over the interval [-1, 2] using the trapezoidal rule with 3 trapezoids of equal height. Our step-by-step calculation yielded an approximate area of 17/2. The trapezoidal rule is a valuable numerical integration technique that allows us to approximate definite integrals when analytical solutions are difficult or impossible to obtain. By dividing the area under the curve into trapezoids and summing their areas, we can achieve a reasonable approximation of the integral. However, it is important to remember that the trapezoidal rule provides an approximation, and the accuracy of the approximation depends on the number of trapezoids used and the nature of the function. Increasing the number of trapezoids generally improves the accuracy, but it also increases the computational effort.

Throughout our exploration, we have highlighted the importance of understanding the underlying concepts of the trapezoidal rule and the steps involved in applying it. We began by understanding the trapezoidal rule, breaking down the formula and its components. This foundational knowledge is crucial for effectively applying the trapezoidal rule to various functions and intervals. We then meticulously applied the trapezoidal rule to our specific function, y = 3 + 2x - x², over the interval [-1, 2], using 3 trapezoids. This involved calculating the width of each trapezoid, identifying the x-coordinates at which to evaluate the function, calculating the corresponding y-values, and finally, plugging these values into the trapezoidal rule formula. By following this step-by-step process, we ensured accuracy and clarity in our estimation.

We further emphasized the importance of a methodical approach by breaking down the calculation process into clear, manageable steps. This included determining the width of each trapezoid, identifying the x-coordinates, calculating the corresponding y-values, and applying the trapezoidal rule formula. Each step was carefully explained and executed, reinforcing the understanding of the trapezoidal rule and its application. This methodical approach is essential for avoiding errors and ensuring the reliability of the approximation. Finally, we analyzed the result, discussing the potential for error, comparing it with other approximation methods, and interpreting it in the context of the problem. This critical analysis provides a deeper understanding of the approximation and its limitations.

In conclusion, the trapezoidal rule is a powerful tool for approximating definite integrals, and our exploration has demonstrated its practical application. By understanding the concepts, following a step-by-step process, and analyzing the results, we can effectively use the trapezoidal rule to estimate areas under curves and solve various problems in calculus and related fields. This article has provided a comprehensive guide to applying the trapezoidal rule, empowering readers to confidently use this technique in their own mathematical endeavors.