y = ½ x^2 – x +3 for 1≤x≤6
(b) Calculate the mid- ordinates for 5 strips between x = 1 and x= 6, and hence use the mid-ordinate rule to approximate the area under the curve between x = 1, x = 6 and the x = axis.
(c) assuming that the area determined by integration to be the actual area, calculate the percentage error in using the mid-ordinate rule.
mmhh
Did you not go back and check your post when you posted this question a few days back?
http://www.jiskha.com/display.cgi?id=1380495416
To calculate the mid-ordinates for 5 strips between x = 1 and x = 6 and use the mid-ordinate rule to approximate the area under the curve, follow these steps:
Step 1: Determine the width of each strip.
The width of each strip is given by the formula:
Width = (end point - start point) / Number of strips
In this case, the end point is 6, the start point is 1, and the number of strips is 5.
Width = (6 - 1) / 5 = 1
Step 2: Determine the x-values for the mid-ordinates.
The x-values for the mid-ordinates can be found by adding half of the width to each start point of each strip.
The x-values are: 1.5, 2.5, 3.5, 4.5, 5.5
Step 3: Substitute the x-values into the equation to find the corresponding y-values.
Substitute each x-value from Step 2 into the equation y = ½ x^2 - x + 3 to calculate the y-values.
For example, when x = 1.5:
y = ½ * (1.5)^2 - 1.5 + 3 = 3.375
Repeat this calculation for each x-value to find the corresponding y-values:
x = 1.5, y = 3.375
x = 2.5, y = 4.625
x = 3.5, y = 5.125
x = 4.5, y = 5.375
x = 5.5, y = 5.375
Step 4: Calculate the sum of the y-values.
Add up all the calculated y-values:
3.375 + 4.625 + 5.125 + 5.375 + 5.375 = 24.875
Step 5: Calculate the area under the curve using the mid-ordinate rule.
The area of each strip is the width multiplied by the average of the two adjacent mid-ordinates.
The average of the y-values for each strip pair is calculated as follows:
(3.375 + 4.625) / 2 = 4
(4.625 + 5.125) / 2 = 4.875
(5.125 + 5.375) / 2 = 5.25
(5.375 + 5.375) / 2 = 5.375
Now we can calculate the area under the curve using the mid-ordinate rule:
Area = Width * (Sum of all average y-values)
Area = 1 * (4 + 4.875 + 5.25 + 5.375 + 5.375) = 24.875 square units
Next, to calculate the percentage error in using the mid-ordinate rule, follow these steps:
Step 1: Calculate the actual area using integration.
To find the actual area under the curve, integrate the equation y = ½ x^2 - x + 3 from x = 1 to x = 6.
The integral of y = ½ x^2 - x + 3 can be calculated as follows:
Area = ∫[1,6] (½ x^2 - x + 3) dx
Area = (1/6)*[x^3 - 3x^2 + 6x] from 1 to 6
Area = (1/6)*[(6^3 - 3*6^2 + 6*6) - (1^3 - 3*1^2 + 6*1)]
Step 2: Calculate the percentage error.
Percentage Error = |(Approximated Area - Actual Area) / Actual Area| * 100
Percentage Error = |(24.875 - Actual Area) / Actual Area| * 100
Substitute the calculated values from Steps 1 and 5 to find the percentage error in using the mid-ordinate rule.