Given f (x) = √(7 + x) + 4 . Use the rectangle method to approximate the area on the interval [0, 8] using 4
rectangles. (Assume the graph goes through the midpoint of each rectangle.)
To approximate the area under the curve using the rectangle method, follow these steps:
1. Divide the interval [0, 8] into four equal subintervals by finding the width of each subinterval:
Width = (b - a) / n,
where b is the upper limit (8), a is the lower limit (0), and n is the number of subintervals (4).
Width = (8 - 0) / 4 = 2
So, each subinterval has a width of 2.
2. Determine the x-values for the left endpoints of each subinterval:
Starting from the leftmost boundary (0), add the width to each subsequent value to find the left endpoints.
x0 = 0
x1 = x0 + Width = 0 + 2 = 2
x2 = x1 + Width = 2 + 2 = 4
x3 = x2 + Width = 4 + 2 = 6
x4 = x3 + Width = 6 + 2 = 8
This gives the x-values for the left endpoints of each subinterval: 0, 2, 4, 6, and 8.
3. Evaluate the function f(x) at each left endpoint to find the corresponding y-values:
y0 = f(0) = √(7 + 0) + 4 = √7 + 4
y1 = f(2) = √(7 + 2) + 4 = √9 + 4 = 3 + 4 = 7
y2 = f(4) = √(7 + 4) + 4 = √11 + 4
y3 = f(6) = √(7 + 6) + 4 = √13 + 4
y4 = f(8) = √(7 + 8) + 4 = √15 + 4
These are the corresponding y-values for each subinterval.
4. Calculate the area of each rectangle:
The area of each rectangle is given by the width (2) multiplied by the height (y-value of each subinterval).
Area0 = Width * y0 = 2 * (√7 + 4)
Area1 = Width * y1 = 2 * 7
Area2 = Width * y2 = 2 * (√11 + 4)
Area3 = Width * y3 = 2 * (√13 + 4)
Area4 = Width * y4 = 2 * (√15 + 4)
5. Add up the areas of all four rectangles to get the approximate total area:
Total Area = Area0 + Area1 + Area2 + Area3 + Area4
Substitute the values of the areas into the equation to get the final answer.
That's how you can use the rectangle method to approximate the area on the interval [0, 8] using 4 rectangles.