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The sum of first and last ordinates, add twice the sum of the remaining odd ordinates and four times the sum of all the even ordinates. The total sum thus obtained is multiplied by one-third of the common distance between the ordinates and the result gives the required area. This rule of finding the area is called:
- a)Mid-ordinate rule
- b)Trapezoidal rule
- c)Average ordinate rule
- d)Simpson's rule
Correct answer is option 'D'. Can you explain this answer?
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Most Upvoted Answer
The sum of first and last ordinates, add twice the sum of the remainin...
Explanation:
The rule of finding the area described in the question is called Simpson's rule. This rule is used to approximate the area under a curve by dividing it into a series of parabolic segments.
Steps of Simpson's Rule:
1. Sum of first and last ordinates: The first step is to find the sum of the first and last ordinates. This means adding the y-values of the first and last points on the curve.
2. Twice the sum of the remaining odd ordinates: Next, we calculate twice the sum of the remaining odd ordinates. This involves adding together the y-values of all the odd points on the curve, excluding the first and last points.
3. Four times the sum of all the even ordinates: We then calculate four times the sum of all the even ordinates. This means adding together the y-values of all the even points on the curve.
4. Total sum: The total sum obtained from step 1, step 2, and step 3 is calculated.
5. Common distance between the ordinates: We multiply the total sum by one-third of the common distance between the ordinates. The common distance refers to the x-coordinate spacing between each point on the curve.
6. Result: The result of step 5 gives the required area under the curve.
Advantages of Simpson's Rule:
- Simpson's rule provides a more accurate approximation of the area under a curve compared to other methods, such as the trapezoidal rule or the mid-ordinate rule.
- It takes into account the curvature of the curve by using parabolic segments, which provides a better fit to the actual shape of the curve.
- The rule is relatively simple to apply and can be used for a wide range of curve shapes.
Conclusion:
Simpson's rule is a method used in numerical integration to calculate the area under a curve. It provides a more accurate estimation compared to other methods and takes into account the curvature of the curve. By following the steps outlined above, the required area can be obtained using Simpson's rule.
The rule of finding the area described in the question is called Simpson's rule. This rule is used to approximate the area under a curve by dividing it into a series of parabolic segments.
Steps of Simpson's Rule:
1. Sum of first and last ordinates: The first step is to find the sum of the first and last ordinates. This means adding the y-values of the first and last points on the curve.
2. Twice the sum of the remaining odd ordinates: Next, we calculate twice the sum of the remaining odd ordinates. This involves adding together the y-values of all the odd points on the curve, excluding the first and last points.
3. Four times the sum of all the even ordinates: We then calculate four times the sum of all the even ordinates. This means adding together the y-values of all the even points on the curve.
4. Total sum: The total sum obtained from step 1, step 2, and step 3 is calculated.
5. Common distance between the ordinates: We multiply the total sum by one-third of the common distance between the ordinates. The common distance refers to the x-coordinate spacing between each point on the curve.
6. Result: The result of step 5 gives the required area under the curve.
Advantages of Simpson's Rule:
- Simpson's rule provides a more accurate approximation of the area under a curve compared to other methods, such as the trapezoidal rule or the mid-ordinate rule.
- It takes into account the curvature of the curve by using parabolic segments, which provides a better fit to the actual shape of the curve.
- The rule is relatively simple to apply and can be used for a wide range of curve shapes.
Conclusion:
Simpson's rule is a method used in numerical integration to calculate the area under a curve. It provides a more accurate estimation compared to other methods and takes into account the curvature of the curve. By following the steps outlined above, the required area can be obtained using Simpson's rule.
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Community Answer
The sum of first and last ordinates, add twice the sum of the remainin...
Simpson's rule:
This rule is based on the assumption that the figures are trapezoids.
In order to apply Simpson's rule, the area must be divided in even number i.e., the number of offsets must be odd i.e., n term in the last offset 'On' should be odd.
The area is given by Simpson's rule:
Important Points
This rule is based on the assumption that the figures are trapezoids.
In order to apply Simpson's rule, the area must be divided in even number i.e., the number of offsets must be odd i.e., n term in the last offset 'On' should be odd.
The area is given by Simpson's rule:
Important Points
- In case of an even number of cross-sections, the end strip is treated separately and the area of the remaining strip is calculated by Simpson's rule. The area of the last strip can be calculated by either trapezoidal or Simpson's rule.
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The sum of first and last ordinates, add twice the sum of the remaining odd ordinates and four times the sum of all the even ordinates. The total sum thus obtained is multiplied by one-third of the common distance between the ordinates and the result gives the required area. This rule of finding the area is called:a)Mid-ordinate ruleb)Trapezoidal rulec)Average ordinate ruled)Simpsons ruleCorrect answer is option 'D'. Can you explain this answer?
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