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Which of the following indicates the assumption assumed in the trapezoidal formula?
- a)mid-area is the mean of the starting area
- b)mid-area is the mean of the end area
- c)mid-area is the mean
- d)mid-area is not the mean of the end area
Correct answer is option 'B'. Can you explain this answer?
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Which of the following indicates the assumption assumed in the trapezo...
Understanding the Trapezoidal Formula
The trapezoidal rule is a numerical method for approximating the definite integral of a function. It is particularly useful in civil engineering for calculating areas under curves. The assumption in the trapezoidal formula is crucial for its application.
Key Assumption in the Trapezoidal Formula
- The trapezoidal rule assumes that the function being evaluated is linear between the two endpoints of the interval.
- This means that the area of the trapezoid formed by connecting the endpoints of the function at the interval can be accurately represented.
Why Option B is Correct
- Option B states: "mid-area is the mean of the end area." This is accurate because the trapezoidal rule estimates the area under the curve by averaging the function values at the two endpoints.
- The formula for the trapezoidal rule is:
\[
\text{Area} \approx \frac{(b-a)}{2} \times [f(a) + f(b)]
\]
where \(f(a)\) and \(f(b)\) are the values of the function at the endpoints \(a\) and \(b\).
- The assumption is that the average of these two function values (the heights of the trapezoid) represents the mid-area correctly, making it reasonable to take the average of the end areas.
Implications of the Assumption
- If the function is not linear, the trapezoidal rule may lead to underestimation or overestimation of the actual area.
- Understanding this assumption helps engineers apply the trapezoidal rule more effectively in real-world scenarios, ensuring more accurate calculations in project designs and assessments.
This detailed understanding of the trapezoidal formula and its assumptions enhances the reliability of numerical integration in civil engineering applications.
The trapezoidal rule is a numerical method for approximating the definite integral of a function. It is particularly useful in civil engineering for calculating areas under curves. The assumption in the trapezoidal formula is crucial for its application.
Key Assumption in the Trapezoidal Formula
- The trapezoidal rule assumes that the function being evaluated is linear between the two endpoints of the interval.
- This means that the area of the trapezoid formed by connecting the endpoints of the function at the interval can be accurately represented.
Why Option B is Correct
- Option B states: "mid-area is the mean of the end area." This is accurate because the trapezoidal rule estimates the area under the curve by averaging the function values at the two endpoints.
- The formula for the trapezoidal rule is:
\[
\text{Area} \approx \frac{(b-a)}{2} \times [f(a) + f(b)]
\]
where \(f(a)\) and \(f(b)\) are the values of the function at the endpoints \(a\) and \(b\).
- The assumption is that the average of these two function values (the heights of the trapezoid) represents the mid-area correctly, making it reasonable to take the average of the end areas.
Implications of the Assumption
- If the function is not linear, the trapezoidal rule may lead to underestimation or overestimation of the actual area.
- Understanding this assumption helps engineers apply the trapezoidal rule more effectively in real-world scenarios, ensuring more accurate calculations in project designs and assessments.
This detailed understanding of the trapezoidal formula and its assumptions enhances the reliability of numerical integration in civil engineering applications.
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Which of the following indicates the assumption assumed in the trapezo...
Trapezoidal formula is based on the assumption that the mid-area is the mean of the end area. Based on this, the trapezoidal formula will be worked out and further calculations are done.
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Which of the following indicates the assumption assumed in the trapezoidal formula?a)mid-area is the mean of the starting areab)mid-area is the mean of the end areac)mid-area is the meand)mid-area is not the mean of the end areaCorrect answer is option 'B'. Can you explain this answer?
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Which of the following indicates the assumption assumed in the trapezoidal formula?a)mid-area is the mean of the starting areab)mid-area is the mean of the end areac)mid-area is the meand)mid-area is not the mean of the end areaCorrect answer is option 'B'. Can you explain this answer? for Civil Engineering (CE) 2025 is part of Civil Engineering (CE) preparation. The Question and answers have been prepared according to the Civil Engineering (CE) exam syllabus. Information about Which of the following indicates the assumption assumed in the trapezoidal formula?a)mid-area is the mean of the starting areab)mid-area is the mean of the end areac)mid-area is the meand)mid-area is not the mean of the end areaCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Civil Engineering (CE) 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which of the following indicates the assumption assumed in the trapezoidal formula?a)mid-area is the mean of the starting areab)mid-area is the mean of the end areac)mid-area is the meand)mid-area is not the mean of the end areaCorrect answer is option 'B'. Can you explain this answer?.
Which of the following indicates the assumption assumed in the trapezoidal formula?a)mid-area is the mean of the starting areab)mid-area is the mean of the end areac)mid-area is the meand)mid-area is not the mean of the end areaCorrect answer is option 'B'. Can you explain this answer? for Civil Engineering (CE) 2025 is part of Civil Engineering (CE) preparation. The Question and answers have been prepared according to the Civil Engineering (CE) exam syllabus. Information about Which of the following indicates the assumption assumed in the trapezoidal formula?a)mid-area is the mean of the starting areab)mid-area is the mean of the end areac)mid-area is the meand)mid-area is not the mean of the end areaCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Civil Engineering (CE) 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which of the following indicates the assumption assumed in the trapezoidal formula?a)mid-area is the mean of the starting areab)mid-area is the mean of the end areac)mid-area is the meand)mid-area is not the mean of the end areaCorrect answer is option 'B'. Can you explain this answer?.
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Which of the following indicates the assumption assumed in the trapezoidal formula?a)mid-area is the mean of the starting areab)mid-area is the mean of the end areac)mid-area is the meand)mid-area is not the mean of the end areaCorrect answer is option 'B'. Can you explain this answer?, a detailed solution for Which of the following indicates the assumption assumed in the trapezoidal formula?a)mid-area is the mean of the starting areab)mid-area is the mean of the end areac)mid-area is the meand)mid-area is not the mean of the end areaCorrect answer is option 'B'. Can you explain this answer? has been provided alongside types of Which of the following indicates the assumption assumed in the trapezoidal formula?a)mid-area is the mean of the starting areab)mid-area is the mean of the end areac)mid-area is the meand)mid-area is not the mean of the end areaCorrect answer is option 'B'. Can you explain this answer? theory, EduRev gives you an
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