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.
I will assume that you are calculating the heights at
x = 1.5, 2.5, 3.5, 4.5, and 5.5
which would be :
2.625, 3.625, 5.625, 8.625, and 12.625
approximate area of the 5 parts
= (2.625)(1) + (3.625)(1) + (5.625)(1) + (8.625)(1) + (12.625)(1)
= 33.125
actual area
= ∫((1/2)x^2 - x + 3) dx from x = 1 to 6
= [ (1/6)x^3 - x^2/2 + 3x ] from x = 1 to 6
= (1/6)(216) - 36/2 + 18 -(3/2 - 1/2 + 3)
= 36 - 18 + 18 - 3/2 + 1/2 - 3
= 32
error = 33.125-32 = 1.125
percentage error = 1.125/32*100% = appr 3.5 %
To get the answer, we will follow these steps:
(b) 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 between x = 1, x = 6, and the x-axis.
Step 1: Determine the width of the strips.
Since we have 5 strips between x = 1 and x = 6, the width of each strip would be:
Strip width = (x_max - x_min) / number of strips
= (6 - 1) / 5
= 1
Step 2: Calculate the mid-ordinates for each strip.
We will calculate the y-coordinate at the midpoint of each strip by substituting the x-coordinate of the midpoint into the equation y = 1/2 x^2 – x + 3.
For the first strip:
Midpoint (x-coordinate) = (x_min + strip width/2)
= 1 + 1/2
= 1.5
Substituting x = 1.5 into the equation, we get:
y = 1/2(1.5)^2 - 1.5 + 3
= 0.375
Similarly, calculate the mid-ordinates for the remaining 4 strips using the same approach:
2nd strip: x = 2.5
3rd strip: x = 3.5
4th strip: x = 4.5
5th strip: x = 5.5
Step 3: Calculate the area under each strip.
The area under each strip is given by the product of the strip width and the corresponding mid-ordinate.
1st strip: Area = strip width * mid-ordinate
= 1 * 0.375
= 0.375
Similarly, calculate the areas for the remaining 4 strips using the same approach.
Step 4: Approximate the total area under the curve.
To approximate the total area under the curve, add up the areas of all the strips.
Total area ≈ Sum of areas of all strips
(c) To calculate the percentage error in using the mid-ordinate rule, we will compare it to the actual area determined by integration.
Step 1: Use integration to find the actual area under the curve between x = 1 and x = 6.
The actual area under the curve can be calculated by integrating the function y = 1/2 x^2 – x + 3 with respect to x over the interval [1, 6].
Step 2: Calculate the actual area using integration.
Step 3: Calculate the percentage error.
To calculate the percentage error, use the formula:
Percentage error = [(approximated area - actual area) / actual area] * 100
Substitute the values of the approximated and actual areas into the formula to calculate the percentage error.
Note: Make sure to use the appropriate precision and rounding rules for the calculations to obtain accurate results.