Use the mid-point rule with n = 4 to approximate the area of the region bounded by y = -x^3 and y = -x
algebraically, the area is zero, since both functions are odd.
geometrically, we can use symmetry and just take twice the area in the 2nd quadrant.
That means we just need two rectangles, with boundaries at
x = 0, -1/2, -1
so the area is twice
(f(0)+f(-1/2))/2 * 1/2 + (f(-1/2)+f(-1))/2 * 1/2
To use the mid-point rule to approximate the area of a region bounded by two curves, we need to divide the region into equal intervals and find the midpoint of each interval. Then, we evaluate the functions at these midpoints to find the height of rectangles that will approximate the area between the curves. Finally, we calculate the sum of these areas to get the total approximation.
In this case, we have the region bounded by y = -x^3 and y = -x. To approximate the area using the mid-point rule with n = 4, we need to divide the interval into four equal subintervals.
Step 1: Find the interval of x-values
To find the interval of x-values, we need to determine the points where the two curves intersect. In this case, -x^3 = -x, so we solve for x:
-x^3 = -x
x^3 - x = 0
x(x^2 - 1) = 0
x = 0 or x = ±1
So, the interval of x-values is [-1, 1].
Step 2: Divide the interval into subintervals
Since n = 4, we need to divide the interval [-1, 1] into 4 equal subintervals. Each subinterval will have a width of (b - a) / n = (1 - (-1)) / 4 = 2 / 4 = 0.5. Therefore, the subintervals are [-1, -0.5], [-0.5, 0], [0, 0.5], and [0.5, 1].
Step 3: Find the midpoints of each subinterval
To find the midpoint of each subinterval, we take the average of the endpoints. For example, the midpoint of the first subinterval is (-1 + (-0.5)) / 2 = -0.75. Similarly, the midpoints of the other subintervals are -0.25, 0.25, and 0.75.
Step 4: Evaluate the functions at the midpoints
Now, we evaluate the functions y = -x^3 and y = -x at each midpoint to find the height of the rectangles.
For the first subinterval [-1, -0.5]:
-(-0.75^3) = -0.421875
-(-0.75) = 0.75
For the second subinterval [-0.5, 0]:
-(0.25^3) = -0.015625
-(0.25) = -0.25
For the third subinterval [0, 0.5]:
-(0.25^3) = -0.015625
-(0.25) = -0.25
For the fourth subinterval [0.5, 1]:
-(0.75^3) = -0.421875
-(0.75) = -0.75
Step 5: Calculate the sum of the areas
The width of each rectangle is 0.5, which is the same for all rectangles.
So, the area of each rectangle is the height multiplied by the width.
For the first subinterval: area = 0.421875 * 0.5 = 0.2109375
For the second subinterval: area = 0.015625 * 0.5 = 0.0078125
For the third subinterval: area = 0.015625 * 0.5 = 0.0078125
For the fourth subinterval: area = 0.421875 * 0.5 = 0.2109375
Finally, we sum up the areas of all four rectangles:
Total approximate area = 0.2109375 + 0.0078125 + 0.0078125 + 0.2109375 = 0.4375
Therefore, using the mid-point rule with n = 4, the approximate area of the region bounded by y = -x^3 and y = -x is 0.4375 square units.