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.)
(sigma)M = (b-a) f((a+b)/2)
width:
(b-a)=2-0
(b-a)=4-2
(b-a)=6-4
(b-a)=8-6 =2
find the height:
f((a+b)/2)
(e.g. ([√(7 + a) + 4] + [√(7 + b) + 4])/2)
take the sum of "rectangles" (i.e. product of width and height)
rectangle method:
http://spiff.rit.edu/classes/phys317/lectures/num_integ2/num_integ2.html
To approximate the area using the rectangle method, we divide the interval [0, 8] into 4 equal sub-intervals.
Step 1: Determine the width of each rectangle
The total width of the interval is 8 - 0 = 8. Since we are dividing it into 4 equal sub-intervals, the width of each rectangle is 8/4 = 2.
Step 2: Determine the height of each rectangle
To find the height of each rectangle, we need to evaluate the function f(x) = √(7 + x) + 4 at the midpoint of each sub-interval.
For the first rectangle, the midpoint of the sub-interval [0, 2] is (0 + 2)/2 = 1. So, we evaluate f(1) = √(7 + 1) + 4.
For the second rectangle, the midpoint of the sub-interval [2, 4] is (2 + 4)/2 = 3. So, we evaluate f(3) = √(7 + 3) + 4.
For the third rectangle, the midpoint of the sub-interval [4, 6] is (4 + 6)/2 = 5. So, we evaluate f(5) = √(7 + 5) + 4.
For the fourth rectangle, the midpoint of the sub-interval [6, 8] is (6 + 8)/2 = 7. So, we evaluate f(7) = √(7 + 7) + 4.
Step 3: Calculate the area of each rectangle
The area of each rectangle is given by width * height. In this case, the width is 2 (calculated in step 1).
For the first rectangle, the height is f(1) = √8 + 4.
For the second rectangle, the height is f(3) = √10 + 4.
For the third rectangle, the height is f(5) = √12 + 4.
For the fourth rectangle, the height is f(7) = √14 + 4.
Step 4: Add up the areas of all rectangles
To approximate the total area, we calculate the sum of the areas of all four rectangles:
Area ≈ (width * height of first rectangle) + (width * height of second rectangle) + (width * height of third rectangle) + (width * height of fourth rectangle)
Area ≈ (2 * (√8 + 4)) + (2 * (√10 + 4)) + (2 * (√12 + 4)) + (2 * (√14 + 4))
Simply the expression to get the final approximation.