Use the mid-point rule with n = 2 to approximate the area of the region bounded by y equals the cube root of the quantity 16 minus x cubed y = x, and x = 0.
(x) = (16-x^3)^(1/3) - x
the partitions are (0,1) U (1,2)
the midpoints are at X1 = 0.5 and x2=1.5
SUM of deltaX * f(Xi) =
1 * f(0.5) + 1*f(1.5) =
f(0.5) + f(1.5) =
2.0132628476567395905687034358119 + 0.828504754
= 2.841767602
To approximate the area using the midpoint rule, we first need to divide the interval [0, 2] into equal subintervals.
Given n = 2, we will divide the interval into 2 subintervals, each having a width of (2 - 0)/2 = 1.
Next, we need to compute the midpoint of each subinterval. Since we have 2 subintervals, we will have 2 midpoints.
The midpoint of each subinterval is calculated as follows:
- For the first subinterval: (0 + (1/2) * 1) = 0.5
- For the second subinterval: (1 + (1/2) * 1) = 1.5
Now, we evaluate the function at each midpoint to find the corresponding function values.
For the first subinterval (0 - 1):
- At x = 0.5, y = (16 - (0.5)^3)^(1/3) = 2
For the second subinterval (1 - 2):
- At x = 1.5, y = (16 - (1.5)^3)^(1/3) = 1.059
Now, we can calculate the sum of the function values and multiply it by the width of each subinterval to get an approximation of the area between the curves.
Area ≈ (1 * f(0.5)) + (1 * f(1.5))
= 1 * 2 + 1 * 1.059
= 2 + 1.059
= 3.059
Therefore, the approximate area of the region bounded by the given curves using the midpoint rule with n = 2 is 3.059 square units.