Use the mid ordinate rule with strips to find the area bounded by the curve y = x^2 + 1, x = -4, x = 8 and x-axis
How many strips ?
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
To find the area bounded by the curve y = x^2 + 1, the x-axis, and the lines x = -4 and x = 8 using the mid-ordinate rule with strips, follow these steps:
Step 1: Determine the width of each strip.
The width of each strip is equal to the difference between the x-values of adjacent mid-ordinates. In this case, since the lines x = -4 and x = 8 create the boundary, the width of each strip is given by:
Δx = (upper limit - lower limit) / number of strips
Let's say we want to divide the interval [-4, 8] into n strips. Then, the width of each strip is:
Δx = (8 - (-4)) / n
= 12 / n
Step 2: Find the mid-ordinate of each strip.
For each strip, find the x-value at the center of the strip. The mid-ordinate is given by:
x_mid = (x_i + x_{i+1}) / 2
where x_i and x_{i+1} are the x-values of the strip's boundary.
Step 3: Calculate the y-value for each mid-ordinate.
Plug each x_mid value into the equation y = x^2 + 1 to get the corresponding y-value.
Step 4: Calculate the area of each strip.
The area of each strip is given by:
A = width * height
= Δx * y_mid
Step 5: Sum up the areas of all the strips.
Add up the areas of all the individual strips to find the total area bounded by the curve.
For example, let's say we want to use 4 strips.
Calculate the width of each strip:
Δx = (8 - (-4)) / 4
= 3
Find the mid-ordinate for each strip:
Strip 1: x_mid = (-4 + (-1)) / 2 = -2.5
Strip 2: x_mid = (-1 + 2) / 2 = 0.5
Strip 3: x_mid = (2 + 5) / 2 = 3.5
Strip 4: x_mid = (5 + 8) / 2 = 6.5
Calculate the y-values for each mid-ordinate:
y_1 = (-2.5)^2 + 1
y_2 = (0.5)^2 + 1
y_3 = (3.5)^2 + 1
y_4 = (6.5)^2 + 1
Calculate the area of each strip:
A_1 = 3 * y_1
A_2 = 3 * y_2
A_3 = 3 * y_3
A_4 = 3 * y_4
Finally, sum up the areas of all the strips:
Total area = A_1 + A_2 + A_3 + A_4
So, by using the mid-ordinate rule with 4 strips, you can find the area bounded by the curve y = x^2 + 1, x = -4, x = 8, and the x-axis.