Graph
y = ½ x^2 – x +3 for 0≤x≤6
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. assuming that the area determined by integration to be the actual area, calculate the percentage error in using the mid-ordinate rule.
To calculate the mid-ordinates, we need to divide the interval between x = 1 and x = 6 into 5 equal strips.
Step 1: Find the width of each strip.
The total width of the interval is 6 - 1 = 5. To divide it into 5 equal strips, we can calculate the width of each strip by dividing the total width by the number of strips: 5 / 5 = 1.
Step 2: Find the x-values for the mid-ordinates.
The x-values for the mid-ordinates will be the starting point of each strip plus half of the width. Starting from x = 1, the x-values for the mid-ordinates will be: 1 + 0.5 = 1.5, 1.5 + 1 = 2.5, 2.5 + 1 = 3.5, 3.5 + 1 = 4.5, and 4.5 + 1 = 5.5.
Step 3: Calculate the y-values for the mid-ordinates.
Substitute the x-values we found in step 2 into the equation y = ½ x^2 - x + 3 to find the corresponding y-values:
For x = 1.5, y = ½ (1.5)^2 - 1.5 + 3 = 2.25 - 1.5 + 3 = 3.75.
For x = 2.5, y = ½ (2.5)^2 - 2.5 + 3 = 6.25 - 2.5 + 3 = 6.75.
For x = 3.5, y = ½ (3.5)^2 - 3.5 + 3 = 12.25 - 3.5 + 3 = 11.75.
For x = 4.5, y = ½ (4.5)^2 - 4.5 + 3 = 20.25 - 4.5 + 3 = 18.75.
For x = 5.5, y = ½ (5.5)^2 - 5.5 + 3 = 30.25 - 5.5 + 3 = 27.75.
Step 4: Calculate the area using the mid-ordinate rule.
The mid-ordinate rule states that the area under the curve can be approximated by summing the areas of the rectangles formed by the mid-ordinates and the x-axis.
The width of each strip is 1, so we multiply it by the sum of the y-values we found in step 3 and then add them up:
Area ≈ (3.75 + 6.75 + 11.75 + 18.75 + 27.75) * 1 = 68.
Step 5: Calculate the actual area using integration.
To find the actual area under the curve between x = 1 and x = 6, we need to perform integration.
Integrate the equation y = ½ x^2 - x + 3 with respect to x over the interval [1, 6]:
∫[1, 6] (½ x^2 - x + 3) dx = [(1/6) x^3 - (1/2) x^2 + 3x] evaluated from 1 to 6.
[(1/6)(6)^3 - (1/2)(6)^2 + 3(6)] - [(1/6)(1)^3 - (1/2)(1)^2 + 3(1)] =
[(216/6) - (18/2) + 18] - [(1/6) - (1/2) + 3] =
[(36) - (9) + 18] - [(1/6) - (1/2) + 18/6] =
45 - (1/6) + (3 - 3) =
45 - (1/6) = 269/6 ≈ 44.833.
Step 6: Calculate the percentage error.
To calculate the percentage error, we can find the difference between the actual area and the approximated area, divide it by the actual area, and then multiply by 100.
Percentage Error = |(Actual Area - Approximated Area) / Actual Area| * 100
= |(44.833 - 68) / 44.833| * 100
= |-23.167 / 44.833| * 100
= 51.56%.
Therefore, the percentage error in using the mid-ordinate rule to approximate the area under the curve is approximately 51.56%.