Use the mid ordinate rule with 6 strips to find the area bounded by the curve y = x^2 + 1, x = -4, x = 8 and x-axis
Your first step would have been to make a sketch, after that it becomes obvious
From -4 to 8 is 12 units, you want 6 strips, so each strip has a width of 2
the midpoints of these strips (the x's) are at
-3, -1, 1, 3, 5, and 7
the corresponding height, (the y values) are
10, 2, 2, 10, 26, and 50
your estimated area = 2(10+2+2+10+26+50) = .....
To use the mid-ordinate rule, we need to divide the region between x = -4 and x = 8 into 6 equal strips. Each strip will have a width of (8 - (-4))/6 = 2 units.
Step 1: Calculate the mid-ordinate for each strip.
We will calculate the y-value of the curve for each x-value in the middle of each strip.
For the first strip:
x = -4 + (1/2) * 2 = -3
y = (-3)^2 + 1 = 9 + 1 = 10
For the second strip:
x = -4 + (3/2) * 2 = -2
y = (-2)^2 + 1 = 4 + 1 = 5
For the third strip:
x = -4 + (5/2) * 2 = 0
y = (0)^2 + 1 = 0 + 1 = 1
For the fourth strip:
x = -4 + (7/2) * 2 = 2
y = (2)^2 + 1 = 4 + 1 = 5
For the fifth strip:
x = -4 + (9/2) * 2 = 4
y = (4)^2 + 1 = 16 + 1 = 17
For the sixth strip:
x = -4 + (11/2) * 2 = 6
y = (6)^2 + 1 = 36 + 1 = 37
Step 2: Calculate the area of each strip.
The area of each strip is equal to the width of the strip (2 units) times the mid-ordinate (y-value).
For the first strip: Area 1 = 2 * 10 = 20 sq.units
For the second strip: Area 2 = 2 * 5 = 10 sq.units
For the third strip: Area 3 = 2 * 1 = 2 sq.units
For the fourth strip: Area 4 = 2 * 5 = 10 sq.units
For the fifth strip: Area 5 = 2 * 17 = 34 sq.units
For the sixth strip: Area 6 = 2 * 37 = 74 sq.units
Step 3: Calculate the total area.
To find the total area, we sum up the areas of all the strips.
Total Area = Area 1 + Area 2 + Area 3 + Area 4 + Area 5 + Area 6
= 20 + 10 + 2 + 10 + 34 + 74
= 150 sq.units
Therefore, the area bounded by the curve y = x^2 + 1, x = -4, x = 8, and the x-axis using the mid-ordinate rule with 6 strips is 150 square units.
To use the mid-ordinate rule with 6 strips to find the area bounded by the curve y = x^2 + 1, x = -4, x = 8, and the x-axis, we need to follow these steps:
Step 1: Determine the width of each strip.
To use 6 strips, we divide the total interval (from x = -4 to x = 8) into 6 equal parts. The width of each strip is given by (8 - (-4)) / 6 = 12/6 = 2.
Step 2: Evaluate the midpoint of each strip.
To find the midpoints, we add half of the strip width to the left endpoint of each strip.
The midpoints for 6 strips are:
(x1, x2, x3, x4, x5, x6) = (-4 + 1, -4 + 3, -4 + 5, -4 + 7, -4 + 9, -4 + 11)
= (-3, -1, 1, 3, 5, 7)
Step 3: Find the corresponding y-values for each midpoint.
Using the equation y = x^2 + 1, substitute the x-values found in step 2 to get the corresponding y-values.
The y-values for the midpoints are:
(y1, y2, y3, y4, y5, y6) = ((-3)^2 + 1, (-1)^2 + 1, (1)^2 + 1, (3)^2 + 1, (5)^2 + 1, (7)^2 + 1)
= (10, 2, 2, 10, 26, 50)
Step 4: Calculate the area of each strip.
For each strip, the area can be approximated by multiplying the width of the strip by the corresponding y-value.
The areas for each strip are:
(A1, A2, A3, A4, A5, A6) = (2 * 10, 2 * 2, 2 * 2, 2 * 10, 2 * 26, 2 * 50)
= (20, 4, 4, 20, 52, 100)
Step 5: Sum up the areas of all strips.
The total area can be found by summing up the areas of all six strips.
Total Area = A1 + A2 + A3 + A4 + A5 + A6
= 20 + 4 + 4 + 20 + 52 + 100
= 200
Therefore, the area bounded by the curve y = x^2 + 1, x = -4, x = 8, and the x-axis using the mid-ordinate rule with 6 strips is 200 square units.