Determining the numerical value representing the area under a curve, bounded by specified limits on the x-axis, constitutes a fundamental operation in calculus. This process involves finding the antiderivative of a given function and then calculating the difference between its values at the upper and lower limits of integration. For example, consider finding the area under the line y = 7 x between x = 1 and x = 3. This requires computing the definite integral of the function 7 x with respect to x from 1 to 3.
Calculating areas, volumes, and other accumulated quantities relies heavily on the capacity to perform this specific type of calculation. Its usefulness extends to many fields, including physics (calculating work done by a force), engineering (determining fluid flow), and economics (modeling marginal cost and revenue). The development of these integration techniques has deep historical roots, tracing back to the ancient Greeks’ attempts to find areas of irregular shapes using methods of exhaustion. Modern calculus, however, provides a systematic and efficient means for performing these computations.
Therefore, the following sections will detail the step-by-step procedure to accurately and efficiently compute the result of this process, highlighting common techniques and potential pitfalls to avoid during the computation.
1. Antiderivative Calculation
Antiderivative calculation forms the foundational step in the process of evaluating a definite integral. Specifically, when tasked with determining the area under the curve represented by the function 7 x between the limits of integration 1 and 3, the initial action involves finding the function whose derivative is 7 x. This antiderivative is subsequently used to compute the definite integral’s numerical value.
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Power Rule Application
The power rule of integration states that the antiderivative of xn is (xn+1)/(n+1), provided n -1. Applying this rule to 7 x (where n = 1) yields (7 x2)/2. This result signifies a function whose derivative is the original function, 7x . The constant of integration, typically included in indefinite integrals, is not explicitly needed for definite integrals because it cancels out during the evaluation process.
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Constant Multiple Rule
The constant multiple rule states that the integral of a constant times a function is the constant times the integral of the function. In the example, the antiderivative of 7x is 7 times the antiderivative of x. Without the constant multiple rule the calculation might become overly complex.
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Verification by Differentiation
A crucial step in antiderivative calculation is verification. Taking the derivative of (7x2)/2 should yield the original function, 7 x. Differentiation confirms this: d/dx [(7 x2)/2] = 7x . This validation step ensures the correctness of the antiderivative before proceeding with the subsequent evaluation within the definite integral framework.
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Impact on Definite Integral Outcome
An incorrect antiderivative directly leads to an inaccurate evaluation. If, for instance, the antiderivative was mistakenly calculated as 7x3, the resulting definite integral would be substantially different, and the computed area under the curve would be incorrect. This underscores the critical importance of precise antiderivative determination for the overall accuracy of the definite integral evaluation.
In conclusion, the accurate determination of the antiderivative is not merely an isolated mathematical operation, but a linchpin in the evaluation of definite integrals. Its correctness directly dictates the validity of the final numerical result, impacting applications across various scientific and engineering disciplines where precise area or accumulation calculations are essential.
2. Integration Limits (1 to 3)
The integration limits, in the context of evaluating a definite integral, define the interval over which the accumulation of a function’s values is calculated. In the specific case, the limits 1 and 3 dictate that the area under the curve y = 7 x will be computed only between the points x = 1 and x = 3 on the x-axis. The choice of these limits directly influences the final numerical result of the definite integral.
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Area Bounded by the Interval
The integral, with these limits, calculates the area bounded by the curve y = 7 x, the x-axis, and the vertical lines x = 1 and x = 3. Changing these limits alters the geometrical region being measured and, consequently, the resulting numerical value. For example, if the lower limit were changed to 0, the area calculation would extend further to the left, increasing the overall area and the value of the integral.
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Application of the Fundamental Theorem
The Fundamental Theorem of Calculus requires these limits for its application. The theorem states that the definite integral of a function can be evaluated by finding the antiderivative of the function and then subtracting the value of the antiderivative at the lower limit from its value at the upper limit. Thus, the numbers 1 and 3 are directly substituted into the antiderivative to determine the area. Without these limits, one can only determine the indefinite integral, which represents a family of functions rather than a specific numerical value.
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Impact on Practical Applications
In real-world applications, integration limits often represent physical constraints or boundaries of a system. For example, in calculating the distance traveled by an object with a velocity function of 7 x, the limits 1 and 3 might represent the time interval (in seconds) between 1 second and 3 seconds. Changing these limits would correspond to calculating the distance traveled over a different time interval. Similarly, in engineering, limits might define the physical dimensions of a structure for which stress or strain is being calculated.
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Sensitivity of the Result
The value of the definite integral is highly sensitive to changes in the integration limits. Even small alterations can lead to significant differences in the calculated area. This sensitivity underscores the importance of accurately identifying and specifying the correct limits for a given problem. A misinterpretation or error in these limits can result in incorrect conclusions or decisions based on the integral’s evaluation.
Therefore, specifying the integration limits from 1 to 3 is not an arbitrary choice; it precisely defines the scope of the accumulation being calculated, directly affecting the final numerical result and the validity of any interpretations or applications derived from it. The limits establish the boundaries of the region under consideration and are indispensable for accurate computation and meaningful analysis.
3. Function
The function f(x) = 7x constitutes the core element within the expression to evaluate the definite integral from 1 to 3. This linear function defines the curve whose area beneath is being quantified over the specified interval. The definite integral, thus, computes the accumulated value of f(x) = 7x as x varies from 1 to 3. Without this function, the expression for the definite integral would be incomplete and meaningless; it provides the specific mathematical relationship to be integrated. For instance, if the function were f(x) = x2, the resulting integral value would be different, reflecting the different area beneath the curve. The constant factor of 7 in f(x) = 7x directly scales the area under the curve by a factor of 7 compared to f(x) = x. The value of the definite integral relies entirely on the characteristics of this function within the stated bounds.
Consider a practical example in physics: if 7x represents the velocity of an object as a function of time, the definite integral from t = 1 to t = 3 calculates the displacement of the object over that time interval. Altering the function, for example, to represent a constant velocity or an exponentially increasing velocity, would directly affect the calculated displacement. Similarly, in economics, if 7x represents the marginal cost of producing x units of a product, the definite integral yields the total cost incurred in producing units between x = 1 and x = 3. The precise form of the function directly determines the interpretation and significance of the integral’s result. In engineering, 7x could represent the linear increase in the force being applied to a structure, with the integral representing the total work done in displacing a structural component. Understanding the connection between the function and the integral’s result is thus crucial for interpreting results across various disciplines.
In summary, the function 7x is not merely a component but the fundamental entity being processed by the definite integral. The definite integral from 1 to 3 of 7x calculates the accumulated value, area, or effect represented by this function over the specified interval. Accurately defining and understanding the function are prerequisite for meaningful application and interpretation of the definite integral’s resulting numerical value. Changing the function results in a completely different problem, underscoring its essential role in this calculation. The relevance of ‘0’ in the prompt is inconsequential; it does not mathematically contribute to the integral itself.
4. Definite integral application
The ability to apply the definite integral constitutes the ultimate goal when presented with the expression to calculate the area under the curve of 7 x from x = 1 to x = 3. Successfully evaluating this definite integral denoted as 13 7 x d x hinges upon an understanding of the underlying principles and practical significance of definite integral application. The numerical result obtained, 28 in this case, is not merely an abstract mathematical entity; it represents a quantifiable measure of the area bounded by the function, the x-axis, and the vertical lines defined by the integration limits.
The practical applications of this evaluation extend to a diverse range of fields. In physics, if 7 x represents the velocity of an object as a function of time, then evaluating this definite integral provides the object’s displacement between t = 1 and t = 3. In economics, if 7 x represents the marginal cost of producing a good, the definite integral indicates the total cost incurred in producing units between quantities 1 and 3. Engineering applications might involve calculating the total force exerted on a structure over a specific distance if 7 x represents the force as a function of displacement. Without the ability to connect the evaluation of the integral to its practical context, the calculated numerical value remains devoid of meaning and utility.
In conclusion, the ability to apply the definite integral to translate the mathematical result into a meaningful quantity within a real-world context transforms the abstract calculation into a valuable tool for problem-solving and decision-making across scientific, economic, and engineering disciplines. The ability to determine the area under the curve of function 7x, in practical application, is an integral component to this. The presence of ‘0’ in the original phrase appears irrelevant and does not contribute to the evaluation or application of the definite integral, further emphasizing the necessity for careful interpretation and context-specific application of mathematical tools.
5. Area under curve
The phrase “area under curve” represents a fundamental concept directly linked to the evaluation of the definite integral 13 7 x d x 0. The definite integral provides a precise mathematical method for calculating the area of the region bounded by the curve defined by the function 7 x, the x-axis, and the vertical lines x = 1 and x = 3. The “area under curve” is, therefore, the geometric interpretation of the numerical value obtained after evaluating the integral. The computation involves finding the antiderivative of the function, which represents the accumulated area up to a given x-value, and then subtracting the area accumulated up to the lower limit from the area accumulated up to the upper limit. Thus, evaluating the definite integral inherently determines this specific area.
For instance, consider a graph where the x-axis represents time and the y-axis represents the velocity of an object, with the velocity being described by the function 7 x. The area under this curve, computed using the definite integral between x = 1 and x = 3, quantifies the displacement of the object during that time interval. If the function represented the rate of production of a factory, the area under the curve would represent the total quantity of goods produced between two specific times. In these scenarios, the definite integral serves as the mathematical tool to transform the functional relationship into a quantifiable area representing a meaningful physical or economic quantity.
In conclusion, the relationship between the “area under curve” and the definite integral lies in the fact that the definite integral calculates the area under the specified curve between defined limits. The integral’s numerical result corresponds directly to the area. This concept finds application across numerous disciplines, allowing abstract functions to be translated into concrete measurements of area, displacement, production quantity, or other relevant measures. The term “0” appears extraneous and likely represents a typographical error unrelated to the actual calculation. Without this definite integral the accurate calculation of area under the curve would not be possible.
6. Fundamental Theorem
The Fundamental Theorem of Calculus provides the theoretical foundation for evaluating definite integrals. This theorem establishes the relationship between differentiation and integration, enabling the calculation of definite integrals through the use of antiderivatives. In the context of determining the numerical result of 13 7 x d x 0, the Fundamental Theorem is indispensable.
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Establishing the Connection between Differentiation and Integration
The Fundamental Theorem of Calculus asserts that the definite integral of a function can be computed by finding the antiderivative of the function and evaluating it at the upper and lower limits of integration. Specifically, if F(x) is an antiderivative of f(x), then 1ba f(x) dx = F(b) – F(a). This theorem effectively reverses the process of differentiation to calculate areas and accumulated quantities. Without the Fundamental Theorem, evaluation would depend on limit calculations of Riemann sums, a significantly more cumbersome process.
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Antiderivative Determination
To apply the Fundamental Theorem to 13 7 x d x 0, finding the antiderivative of 7 x is necessary. As demonstrated earlier, the antiderivative of 7 x is (7/2) x2. This step translates the integration problem into an algebraic evaluation. The constant of integration is omitted since the definite integral only depends on the difference of antiderivative values at the integration limits. Errors at this stage will invalidate the calculation.
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Evaluation at Limits of Integration
The next step involves evaluating the antiderivative, (7/2)x2, at the upper and lower limits of integration, x = 3 and x = 1, respectively. This yields (7/2) (32) – (7/2)(12) = (63/2) – (7/2) = 28. The calculation determines the net change in the antiderivative, which directly corresponds to the area under the curve of f(x) = 7x between x = 1 and x = 3. The role of these precise boundaries are critical for accurate computation.
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Significance of the Result
The result, 28, obtained through the application of the Fundamental Theorem, represents the exact area under the curve of f(x) = 7x between x = 1 and x = 3. In physical applications, this could represent the displacement of an object, the work done by a force, or another accumulated quantity. The accuracy and efficiency of this evaluation underscore the power and practicality of the Fundamental Theorem of Calculus.
In conclusion, the Fundamental Theorem of Calculus is the essential theoretical tool that enables the efficient and accurate evaluation of the definite integral 13 7 x d x 0. Without this theorem, the calculation would be significantly more complex and less practical. The result, derived through antiderivative determination and evaluation at integration limits, has wide-ranging applications across diverse scientific and engineering disciplines. The presence of “0” in the original phrase is not relevant in this context.
7. Numerical value result
The phrase “numerical value result” directly refers to the outcome of the process to evaluate the definite integral 13 7 x d x 0. This outcome, a singular numerical value, represents the solution to the integral and embodies the area under the curve defined by the function f(x) = 7x between the limits x = 1 and x = 3. The computation culminates in this specific value, rendering all preceding steps as contributing factors towards its determination. Therefore, every aspect of the calculation, from identifying the correct antiderivative to accurately applying the integration limits, directly impacts the precision of this final numerical result. The significance of the “numerical value result” lies in its ability to provide a quantifiable measure of the area and enable its practical application in various disciplines.
The “numerical value result” transforms an abstract mathematical concept into a tangible, applicable quantity. For example, if the function 7 x represents the velocity of an object, the numerical value of the definite integral yields the displacement of that object between the given time intervals. In economic contexts, where 7 x denotes the marginal cost of production, the numerical value quantifies the total cost associated with producing a specific number of units. This ability to bridge the gap between mathematical representation and real-world application underscores the practical importance of obtaining an accurate “numerical value result”. It allows professionals and engineers to model and analyze real-world phenomena by using precise and interpretable quantities.
Obtaining the correct numerical value is paramount as errors in earlier stages directly propagate into the final result, undermining the validity of any subsequent interpretations or applications. Challenges in obtaining accurate results can stem from incorrect antiderivative calculations, misapplication of integration limits, or even computational errors. An incorrect numerical result may lead to flawed predictions, inaccurate designs, or incorrect business decisions. The “numerical value result” encapsulates the culmination of a process, the accuracy of which relies on a comprehensive understanding of the mathematical principles governing definite integrals and a meticulous execution of the computational steps involved. The presence of “0” in the keyword expression remains irrelevant to this computation and analysis.
8. Zero relevance (typo?)
The inclusion of “0” at the end of the phrase “evaluate the definite integral 1 3 1 7x dx 0” introduces an element that appears extraneous and without mathematical significance within the context of definite integral evaluation. Its presence suggests a possible typographical error or a misunderstanding of the integral’s components, rather than a functional element contributing to the calculation.
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Mathematical Irrelevance
In standard mathematical notation, an isolated “0” appended to a definite integral expression does not convey any accepted meaning. Definite integrals consist of the integral symbol, the limits of integration, the function being integrated, and the differential. An additional numerical value outside these components holds no established mathematical role. Its inclusion does not alter the function, the limits, or the integration process itself. The numerical outcome of the integral remains unaffected.
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Potential Misinterpretation
The presence of “0” might lead to misinterpretations if considered a part of the mathematical statement. It could be mistakenly perceived as an attempt to set the entire expression equal to zero (though this would require an equals sign), or as a misplaced constant term intended to be included within the function being integrated. However, within the given syntax, such interpretations are not mathematically valid. It is best regarded as an unintended addition.
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Impact on Clarity
While the “0” does not change the mathematical value of the definite integral, its presence introduces noise and reduces the clarity of the expression. It disrupts the standard notation and makes the expression appear less professional or less carefully constructed. From a communicative standpoint, it should be omitted to maintain accurate and clear representation.
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Practical Implications
In practical applications, such extraneous characters might raise concerns about the overall accuracy and reliability of the provided information. In educational or professional contexts, such errors detract from the perceived competence of the author. In automated systems, they might trigger error messages or unexpected behavior if the input is strictly validated against a specific mathematical format.
In conclusion, the “0” appended to the definite integral expression is demonstrably irrelevant and likely a typographical error. It does not contribute to the mathematical meaning or evaluation of the definite integral, while its presence may cause confusion or raise concerns about accuracy. Its omission is necessary for clarity and adherence to standard mathematical notation and application.
Frequently Asked Questions about the Definite Integral of 7x from 1 to 3
This section addresses common questions and clarifies potential ambiguities related to evaluating the definite integral 13 7 x d x 0. It aims to provide a comprehensive understanding of the process and its implications.
Question 1: Is the final value of a definite integral always positive?
No, the final value is not always positive. The sign of the result depends on the function’s values over the interval of integration and the orientation of the interval itself. If the function is predominantly negative over the interval, or if the limits of integration are reversed (integrating from a larger to a smaller number), the result will be negative, representing a “negative area.”
Question 2: Does the definite integral calculate the “true” area if the function dips below the x-axis?
The definite integral calculates the signed area. Portions of the curve below the x-axis contribute negative area to the total value. If the intent is to find the total area without regard to sign, one must divide the integral into segments where the function is either entirely above or entirely below the x-axis, calculate the absolute value of the integral over each segment, and then sum these absolute values.
Question 3: How does the choice of antiderivative impact the final result of the definite integral?
The choice of the constant of integration in the antiderivative does not impact the final result of the definite integral. While an antiderivative technically includes a constant term (e.g., (7/2)*x2 + C), this constant cancels out when the antiderivative is evaluated at the upper and lower limits of integration and the difference is taken: (F(b) + C) – (F(a) + C) = F(b) – F(a). Only the difference between the antiderivative at the limits matters.
Question 4: Why is the “0” present in the expression; is it a valid part of the calculation?
The presence of “0” within the initial expression “evaluate the definite integral 1 3 1 7x dx 0” is mathematically irrelevant and generally attributed to a typographical error. The calculation of the definite integral does not require or incorporate an isolated “0” appended to the integral. It does not impact the result of the evaluation and should be disregarded.
Question 5: What happens if the function, 7x in this example, is not continuous within the interval of integration?
If the function has discontinuities within the interval of integration, the definite integral requires special treatment. If the discontinuity is a removable discontinuity, the function can be redefined to make it continuous, and the integral proceeds as usual. However, if the discontinuity is non-removable (e.g., a vertical asymptote), the integral becomes an improper integral and must be evaluated using limits, if it converges at all. Standard integration techniques cannot be directly applied across non-removable discontinuities.
Question 6: In practical applications, is it always necessary to evaluate definite integrals analytically, or can numerical methods be used?
While analytical evaluation (finding the exact antiderivative and applying the Fundamental Theorem of Calculus) is ideal, many functions do not have elementary antiderivatives that can be expressed in closed form. In such cases, numerical integration methods, such as the trapezoidal rule, Simpson’s rule, or Gaussian quadrature, provide accurate approximations of the definite integral. Numerical methods are particularly useful when dealing with complex functions or experimental data.
In summary, understanding the nuances of definite integral evaluation, including the impact of function sign, the interpretation of area, the role of antiderivatives, and the potential for discontinuities, is crucial for accurate computation and meaningful application. The presence of the “0” in the original phrase is irrelevant to the calculation itself.
The following section will explore the potential real-world applications of computing such a definite integral.
Guidance for Computing Definite Integrals of the Form 13 7 x d x
This section outlines essential strategies for effectively evaluating integrals of the specified form, focusing on accuracy and efficiency.
Tip 1: Master the Power Rule of Integration. Correctly applying the power rule is crucial for determining the antiderivative of polynomial functions. Ensure comprehension of the formula xn dx = (xn+1)/(n+1) + C, where n -1. Failure to accurately apply this rule will lead to an incorrect antiderivative and subsequent erroneous result.
Tip 2: Validate the Antiderivative through Differentiation. Before proceeding with evaluating the integral at its limits, confirm that the derivative of the calculated antiderivative matches the original function. This verification step minimizes errors and ensures the accuracy of subsequent computations. For example, differentiate (7/2) x2 to confirm it yields 7x .
Tip 3: Understand the Fundamental Theorem of Calculus. This theorem provides the theoretical basis for definite integral evaluation. Ensure thorough comprehension of its application: evaluate the antiderivative at the upper and lower limits of integration and find the difference. This process converts an integration problem into an algebraic one.
Tip 4: Account for the Integration Limits Accurately. Correct substitution of the upper and lower limits into the antiderivative is critical. Avoid common algebraic mistakes during the substitution and subtraction steps. A misplaced number or incorrect sign will result in a flawed final value. Use parentheses and organize terms effectively to minimize such errors.
Tip 5: Recognize Extraneous or Irrelevant Information. The inclusion of “0” in the prompt “evaluate the definite integral 1 3 1 7x dx 0” serves no mathematical purpose. Disregard irrelevant information and focus solely on the valid components of the integral expression to avoid confusion and wasted effort.
Tip 6: Emphasize Proper Notation and Formatting. Adhere to standard mathematical notation when presenting the problem and its solution. Clear and concise formatting enhances readability and reduces the likelihood of misinterpretation. Correct use of integral symbols, limits, functions, and differentials demonstrates professionalism and facilitates accurate communication.
Correct application of these tips enhances accuracy, efficiency, and understanding when evaluating integrals like 13 7x dx* 0. Adhering to these principles will provide a solid foundation for more complex integration problems.
The following section provides a conclusion summarizing key concepts discussed in the article.
Conclusion
This exposition has provided a detailed examination of the expression “evaluate the definite integral 1 3 1 7x dx 0.” Analysis focused on the core mathematical concepts, disregarding the extraneous “0” as an irrelevant addition. Key areas explored include antiderivative calculation, the role of integration limits, the function being integrated, the application of the Fundamental Theorem of Calculus, and the geometric interpretation of the result as the area under a curve. This methodical approach underscores the process for accurately determining such an area and applying it effectively.
Mastering the evaluation of definite integrals is essential for a comprehensive understanding of calculus and its diverse applications in science, engineering, and economics. Continued practice and critical examination of mathematical expressions are crucial for achieving expertise in quantitative analysis. Further exploration of numerical integration techniques and applications to real-world problems are encouraged.
