Use the Midpoint Rule with
n = 4
to approximate the area of the region bounded by the graph of the function and the x-axis over the given interval.
f(x) = x^2 + 4x, [0, 4]
To approximate the area of the region bounded by the graph of the function f(x) = x^2 + 4x and the x-axis over the interval [0, 4] using the Midpoint Rule with n = 4, you can follow these steps:
Step 1: Determine the width of each subinterval.
The width of each subinterval is given by the formula:
Δx = (b - a) / n,
where a and b are the endpoints of the interval and n is the number of subintervals. In this case, a = 0, b = 4, and n = 4. So, the width of each subinterval is:
Δx = (4 - 0) / 4 = 1.
Step 2: Determine the midpoint of each subinterval.
To find the midpoint of each subinterval, you can use the formula:
xi = a + (i - 0.5) * Δx,
where xi is the midpoint of the i-th subinterval and i ranges from 1 to n. In this case, the midpoint of each subinterval can be found as follows:
For i = 1:
x1 = 0 + (1 - 0.5) * 1 = 0.5
For i = 2:
x2 = 0 + (2 - 0.5) * 1 = 1.5
For i = 3:
x3 = 0 + (3 - 0.5) * 1 = 2.5
For i = 4:
x4 = 0 + (4 - 0.5) * 1 = 3.5
Step 3: Evaluate the function at each midpoint.
Evaluate the function f(x) = x^2 + 4x at each midpoint calculated in step 2. Substitute the value of xi into the function to get the corresponding y-coordinate for each midpoint.
For i = 1:
f(0.5) = (0.5)^2 + 4(0.5) = 0.25 + 2 = 2.25
For i = 2:
f(1.5) = (1.5)^2 + 4(1.5) = 2.25 + 6 = 8.25
For i = 3:
f(2.5) = (2.5)^2 + 4(2.5) = 6.25 + 10 = 16.25
For i = 4:
f(3.5) = (3.5)^2 + 4(3.5) = 12.25 + 14 = 26.25
Step 4: Calculate the area of each rectangle.
Each rectangle has a width of Δx and a height equal to the corresponding y-coordinate of the midpoint.
For i = 1:
Rectangle 1 area = Δx * f(0.5) = 1 * 2.25 = 2.25
For i = 2:
Rectangle 2 area = Δx * f(1.5) = 1 * 8.25 = 8.25
For i = 3:
Rectangle 3 area = Δx * f(2.5) = 1 * 16.25 = 16.25
For i = 4:
Rectangle 4 area = Δx * f(3.5) = 1 * 26.25 = 26.25
Step 5: Sum up the areas of all the rectangles.
To find the approximate area, sum up the areas of all the rectangles calculated in step 4.
Approximate area = Rectangle 1 area + Rectangle 2 area + Rectangle 3 area + Rectangle 4 area
= 2.25 + 8.25 + 16.25 + 26.25
= 53
Therefore, the approximate area of the region bounded by the graph of f(x) = x^2 + 4x and the x-axis over the interval [0, 4] using the Midpoint Rule with n = 4 is 53 square units.