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L(P, f) is the lower Riemann sum over all partitions on [a, b], then choose the incorrect option.
(a) int a ^ b f(x)dx=lu. b\{L(P, f)\}
(b) integrate f(x) dx from a to b = g, lb\{L(P, f)\}
(c) integrate f(x) dx from a to b = L(P, f)
int a ^ b f(x)dx=lu. b\{L(P, f)\}?
(a) int a ^ b f(x)dx=lu. b\{L(P, f)\}
(b) integrate f(x) dx from a to b = g, lb\{L(P, f)\}
(c) integrate f(x) dx from a to b = L(P, f)
int a ^ b f(x)dx=lu. b\{L(P, f)\}?
Most Upvoted Answer
L(P, f) is the lower Riemann sum over all partitions on [a, b], then c...
Understanding Lower Riemann Sums
Lower Riemann sums, denoted as L(P, f), are a fundamental concept in calculus used to approximate the area under a curve. Here, we analyze the options provided to identify the incorrect statement regarding the integral of a function over an interval.
Key Concepts
- Lower Riemann Sum (L(P, f)): This is calculated using the infimum of the function f(x) over subintervals defined by a partition P of [a, b].
- Integral of f(x): The definite integral from a to b, written as ∫[a, b] f(x)dx, represents the exact area under the curve of f(x) over that interval.
Evaluating the Options
- Option (a): ∫[a, b] f(x)dx = lu {L(P, f)}
This statement is true as the integral equals the least upper bound of the lower sums.
- Option (b): ∫[a, b] f(x)dx = g, lb {L(P, f)}
This option seems ambiguous due to the notation used for "g." If "g" implies the integral, the statement holds true.
- Option (c): ∫[a, b] f(x)dx = L(P, f)
This statement is incorrect. The lower Riemann sum does not equal the integral; rather, it provides a lower approximation.
Conclusion
The incorrect option here is (c), as it misrepresents the relationship between the integral and the lower Riemann sum. The integral represents the exact area, while L(P, f) provides a lower estimate based on partitions of the interval.
Understanding these distinctions is crucial in calculus and helps in grasping the nuances of area approximation techniques.
Lower Riemann sums, denoted as L(P, f), are a fundamental concept in calculus used to approximate the area under a curve. Here, we analyze the options provided to identify the incorrect statement regarding the integral of a function over an interval.
Key Concepts
- Lower Riemann Sum (L(P, f)): This is calculated using the infimum of the function f(x) over subintervals defined by a partition P of [a, b].
- Integral of f(x): The definite integral from a to b, written as ∫[a, b] f(x)dx, represents the exact area under the curve of f(x) over that interval.
Evaluating the Options
- Option (a): ∫[a, b] f(x)dx = lu {L(P, f)}
This statement is true as the integral equals the least upper bound of the lower sums.
- Option (b): ∫[a, b] f(x)dx = g, lb {L(P, f)}
This option seems ambiguous due to the notation used for "g." If "g" implies the integral, the statement holds true.
- Option (c): ∫[a, b] f(x)dx = L(P, f)
This statement is incorrect. The lower Riemann sum does not equal the integral; rather, it provides a lower approximation.
Conclusion
The incorrect option here is (c), as it misrepresents the relationship between the integral and the lower Riemann sum. The integral represents the exact area, while L(P, f) provides a lower estimate based on partitions of the interval.
Understanding these distinctions is crucial in calculus and helps in grasping the nuances of area approximation techniques.
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L(P, f) is the lower Riemann sum over all partitions on [a, b], then choose the incorrect option.(a) int a ^ b f(x)dx=lu. b\{L(P, f)\}(b) integrate f(x) dx from a to b = g, lb\{L(P, f)\}(c) integrate f(x) dx from a to b = L(P, f)int a ^ b f(x)dx=lu. b\{L(P, f)\}?
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L(P, f) is the lower Riemann sum over all partitions on [a, b], then choose the incorrect option.(a) int a ^ b f(x)dx=lu. b\{L(P, f)\}(b) integrate f(x) dx from a to b = g, lb\{L(P, f)\}(c) integrate f(x) dx from a to b = L(P, f)int a ^ b f(x)dx=lu. b\{L(P, f)\}? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about L(P, f) is the lower Riemann sum over all partitions on [a, b], then choose the incorrect option.(a) int a ^ b f(x)dx=lu. b\{L(P, f)\}(b) integrate f(x) dx from a to b = g, lb\{L(P, f)\}(c) integrate f(x) dx from a to b = L(P, f)int a ^ b f(x)dx=lu. b\{L(P, f)\}? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for L(P, f) is the lower Riemann sum over all partitions on [a, b], then choose the incorrect option.(a) int a ^ b f(x)dx=lu. b\{L(P, f)\}(b) integrate f(x) dx from a to b = g, lb\{L(P, f)\}(c) integrate f(x) dx from a to b = L(P, f)int a ^ b f(x)dx=lu. b\{L(P, f)\}?.
L(P, f) is the lower Riemann sum over all partitions on [a, b], then choose the incorrect option.(a) int a ^ b f(x)dx=lu. b\{L(P, f)\}(b) integrate f(x) dx from a to b = g, lb\{L(P, f)\}(c) integrate f(x) dx from a to b = L(P, f)int a ^ b f(x)dx=lu. b\{L(P, f)\}? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about L(P, f) is the lower Riemann sum over all partitions on [a, b], then choose the incorrect option.(a) int a ^ b f(x)dx=lu. b\{L(P, f)\}(b) integrate f(x) dx from a to b = g, lb\{L(P, f)\}(c) integrate f(x) dx from a to b = L(P, f)int a ^ b f(x)dx=lu. b\{L(P, f)\}? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for L(P, f) is the lower Riemann sum over all partitions on [a, b], then choose the incorrect option.(a) int a ^ b f(x)dx=lu. b\{L(P, f)\}(b) integrate f(x) dx from a to b = g, lb\{L(P, f)\}(c) integrate f(x) dx from a to b = L(P, f)int a ^ b f(x)dx=lu. b\{L(P, f)\}?.
Solutions for L(P, f) is the lower Riemann sum over all partitions on [a, b], then choose the incorrect option.(a) int a ^ b f(x)dx=lu. b\{L(P, f)\}(b) integrate f(x) dx from a to b = g, lb\{L(P, f)\}(c) integrate f(x) dx from a to b = L(P, f)int a ^ b f(x)dx=lu. b\{L(P, f)\}? in English & in Hindi are available as part of our courses for UPSC.
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