Graph
y = ½ x^2 – x +3 for 0≤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.
To calculate the mid-ordinates and approximate the area under the curve using the mid-ordinate rule, follow these steps:
Step 1: Calculate the width of each strip.
The width of each strip can be computed by dividing the total width of the interval (x = 6 - x = 1) by the number of strips (5 in this case). So, the width of each strip is (6 - 1) / 5 = 1.
Step 2: Determine the x-values for each mid-ordinate.
Starting from x = 1, add the width of each strip to find the x-values for the mid-ordinates. In this case, since we have 5 strips with a width of 1, the x-values would be 1, 2, 3, 4, and 5.
Step 3: Calculate the y-values (ordinates) for each mid-ordinate.
Substitute the x-values obtained in Step 2 into the equation of the curve (y = ½x^2 - x + 3) to find the corresponding y-values. For example, substituting x = 1 gives y = ½(1)^2 - 1 + 3 = 3.
The calculated mid-ordinates and their corresponding y-values will be as follows:
Mid-ordinate 1: (1, 3)
Mid-ordinate 2: (2, 4)
Mid-ordinate 3: (3, 5.5)
Mid-ordinate 4: (4, 6)
Mid-ordinate 5: (5, 6)
Next, we can use the mid-ordinate rule to approximate the area under the curve between x = 1 and x = 6.
Step 4: Calculate the area of each strip.
To calculate the area of each individual strip, multiply the width of the strip by the corresponding y-value (ordinate). For example, the area of the first strip would be 1 * 3 = 3.
Step 5: Sum up the areas of all strips.
Add up the areas of all the individual strips to get the approximate area under the curve. In this case, the sum of the areas would be:
3 + 4 + 5.5 + 6 + 6 = 24.5
Therefore, the approximate area under the curve between x = 1 and x = 6 is 24.5 square units.
Moving on to part (c), which asks for the percentage error in using the mid-ordinate rule compared to the actual area determined by integration.
Step 6: Calculate the actual area using integration.
To find the actual area under the curve, we need to integrate the equation y = ½x^2 - x + 3 over the interval 0 ≤ x ≤ 6. Integrating this equation yields:
∫[0,6] (½x^2 - x + 3) dx = [1/6 * x^3 - 1/2 * x^2 + 3x] evaluated from x = 0 to x = 6
= (1/6 * 6^3 - 1/2 * 6^2 + 3 * 6) - (1/6 * 0^3 - 1/2 * 0^2 + 3 * 0)
= (72 - 18 + 18) - (0 - 0 + 0)
= 72 square units
The actual area under the curve, determined by integration, is 72 square units.
Step 7: Calculate the percentage error.
To find the percentage error, we first need to determine the absolute difference between the approximate area obtained using the mid-ordinate rule (24.5 square units) and the actual area determined by integration (72 square units). The absolute difference is |72 - 24.5| = 47.5.
Next, we calculate the percentage error by dividing the absolute difference by the actual area and then multiplying by 100. Therefore, the percentage error is (47.5 / 72) * 100 ≈ 65.97%.
Hence, the percentage error in using the mid-ordinate rule to approximate the area under the curve is approximately 65.97%.