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P(0,3), Q(0.5, 4), and R (1,5) are three points on the curve defined by f(x). Numerical integration is carried out using both Trapezoidal rule and Simpson‟s rule within limits x = 0 and x =1 for the curve. The difference between the two results will be.
- a)0
- b)0.25
- c)0.5
- d)1
Correct answer is option 'A'. Can you explain this answer?
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Given information:
- Three points on the curve defined by f(x): P(0,3), Q(0.5, 4), and R (1,5).
- Numerical integration is carried out using both Trapezoidal rule and Simpson's rule.
- Limits of integration: x = 0 to x = 1.
To find:
The difference between the results obtained using Trapezoidal rule and Simpson's rule.
Explanation:
In numerical integration, the Trapezoidal rule and Simpson's rule are used to approximate the definite integral of a function over a given interval.
Trapezoidal Rule:
The Trapezoidal rule approximates the area under the curve by dividing the interval into trapezoids and summing up their areas. The formula for the Trapezoidal rule is:
Trapezoidal rule formula: ∫[a, b] f(x) dx ≈ (b - a) * (f(a) + f(b)) / 2
Simpson's Rule:
Simpson's rule approximates the area under the curve by dividing the interval into small parabolic segments and summing up their areas. The formula for Simpson's rule is:
Simpson's rule formula: ∫[a, b] f(x) dx ≈ (b - a) * (f(a) + 4f((a + b) / 2) + f(b)) / 6
Applying Trapezoidal rule:
Using the given points P(0,3), Q(0.5, 4), and R (1,5), we can estimate the definite integral using the Trapezoidal rule.
∫[0, 1] f(x) dx ≈ (1 - 0) * (f(0) + f(1)) / 2
Since we have three points, we can estimate f(0) as 3 and f(1) as 5.
∫[0, 1] f(x) dx ≈ (1 - 0) * (3 + 5) / 2 = 8/2 = 4
Applying Simpson's rule:
Using the given points P(0,3), Q(0.5, 4), and R (1,5), we can estimate the definite integral using Simpson's rule.
∫[0, 1] f(x) dx ≈ (1 - 0) * (f(0) + 4f(0.5) + f(1)) / 6
∫[0, 1] f(x) dx ≈ (1 - 0) * (3 + 4(4) + 5) / 6 = (1/6) * (3 + 16 + 5) = 24/6 = 4
Conclusion:
The results obtained using Trapezoidal rule and Simpson's rule are both equal to 4. Therefore, the difference between the two results is 0. Hence, the correct answer is option 'A'.
- Three points on the curve defined by f(x): P(0,3), Q(0.5, 4), and R (1,5).
- Numerical integration is carried out using both Trapezoidal rule and Simpson's rule.
- Limits of integration: x = 0 to x = 1.
To find:
The difference between the results obtained using Trapezoidal rule and Simpson's rule.
Explanation:
In numerical integration, the Trapezoidal rule and Simpson's rule are used to approximate the definite integral of a function over a given interval.
Trapezoidal Rule:
The Trapezoidal rule approximates the area under the curve by dividing the interval into trapezoids and summing up their areas. The formula for the Trapezoidal rule is:
Trapezoidal rule formula: ∫[a, b] f(x) dx ≈ (b - a) * (f(a) + f(b)) / 2
Simpson's Rule:
Simpson's rule approximates the area under the curve by dividing the interval into small parabolic segments and summing up their areas. The formula for Simpson's rule is:
Simpson's rule formula: ∫[a, b] f(x) dx ≈ (b - a) * (f(a) + 4f((a + b) / 2) + f(b)) / 6
Applying Trapezoidal rule:
Using the given points P(0,3), Q(0.5, 4), and R (1,5), we can estimate the definite integral using the Trapezoidal rule.
∫[0, 1] f(x) dx ≈ (1 - 0) * (f(0) + f(1)) / 2
Since we have three points, we can estimate f(0) as 3 and f(1) as 5.
∫[0, 1] f(x) dx ≈ (1 - 0) * (3 + 5) / 2 = 8/2 = 4
Applying Simpson's rule:
Using the given points P(0,3), Q(0.5, 4), and R (1,5), we can estimate the definite integral using Simpson's rule.
∫[0, 1] f(x) dx ≈ (1 - 0) * (f(0) + 4f(0.5) + f(1)) / 6
∫[0, 1] f(x) dx ≈ (1 - 0) * (3 + 4(4) + 5) / 6 = (1/6) * (3 + 16 + 5) = 24/6 = 4
Conclusion:
The results obtained using Trapezoidal rule and Simpson's rule are both equal to 4. Therefore, the difference between the two results is 0. Hence, the correct answer is option 'A'.
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P(0,3), Q(0.5, 4), and R (1,5) are three points on the curve defined by f(x). Numerical integration is carried out using both Trapezoidal rule and Simpson‟s rule within limits x = 0 and x =1 for the curve. The difference between the two results will be.a)0b)0.25c)0.5d)1Correct answer is option 'A'. Can you explain this answer?
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P(0,3), Q(0.5, 4), and R (1,5) are three points on the curve defined by f(x). Numerical integration is carried out using both Trapezoidal rule and Simpson‟s rule within limits x = 0 and x =1 for the curve. The difference between the two results will be.a)0b)0.25c)0.5d)1Correct answer is option 'A'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about P(0,3), Q(0.5, 4), and R (1,5) are three points on the curve defined by f(x). Numerical integration is carried out using both Trapezoidal rule and Simpson‟s rule within limits x = 0 and x =1 for the curve. The difference between the two results will be.a)0b)0.25c)0.5d)1Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for P(0,3), Q(0.5, 4), and R (1,5) are three points on the curve defined by f(x). Numerical integration is carried out using both Trapezoidal rule and Simpson‟s rule within limits x = 0 and x =1 for the curve. The difference between the two results will be.a)0b)0.25c)0.5d)1Correct answer is option 'A'. Can you explain this answer?.
P(0,3), Q(0.5, 4), and R (1,5) are three points on the curve defined by f(x). Numerical integration is carried out using both Trapezoidal rule and Simpson‟s rule within limits x = 0 and x =1 for the curve. The difference between the two results will be.a)0b)0.25c)0.5d)1Correct answer is option 'A'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about P(0,3), Q(0.5, 4), and R (1,5) are three points on the curve defined by f(x). Numerical integration is carried out using both Trapezoidal rule and Simpson‟s rule within limits x = 0 and x =1 for the curve. The difference between the two results will be.a)0b)0.25c)0.5d)1Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for P(0,3), Q(0.5, 4), and R (1,5) are three points on the curve defined by f(x). Numerical integration is carried out using both Trapezoidal rule and Simpson‟s rule within limits x = 0 and x =1 for the curve. The difference between the two results will be.a)0b)0.25c)0.5d)1Correct answer is option 'A'. Can you explain this answer?.
Solutions for P(0,3), Q(0.5, 4), and R (1,5) are three points on the curve defined by f(x). Numerical integration is carried out using both Trapezoidal rule and Simpson‟s rule within limits x = 0 and x =1 for the curve. The difference between the two results will be.a)0b)0.25c)0.5d)1Correct answer is option 'A'. Can you explain this answer? in English & in Hindi are available as part of our courses for GATE.
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