CALC 2
Using rectangles each of whose height is given by the value of the function at the midpoint of the rectangle's base (the midpoint rule), estimate the area under the graph of the following function, using first two and then four rectangles. f(x) = 2/x between x=5 and x=9
I did the 4 rectangle part of the question.
You do the easier part with only 2 rectangles
Each rectangle has a width of 1 unit and the midpoint values are
5.5, 6.5, 7.5, and 8.5
f(5.5) = 2/5.5 = .3636
f(6.5) = 2/6.5 = .3077
f(7.5) = .2667
f(8.5) = .2353
appr area = (1)(.3636 + .3077 + .2667 + .2353) = 1.1733
(actual area by calculus) = 1.1756
To estimate the area under the graph of the function f(x) = 2/x between x = 5 and x = 9 using the midpoint rule, we need to divide the interval into rectangles and calculate the area of each rectangle.
First, let's start with two rectangles:
1. Divide the interval [5, 9] into two equal subintervals: [5, 7] and [7, 9].
2. For each subinterval, find the midpoint. The midpoint of [5, 7] is (5 + 7)/2 = 6, and the midpoint of [7, 9] is (7 + 9)/2 = 8.
3. Calculate the height of each rectangle by evaluating the function at the corresponding midpoint. For the first rectangle, the height is f(6) = 2/6, and for the second rectangle, the height is f(8) = 2/8.
4. Calculate the width of each rectangle. Since we have divided the interval into two equal subintervals, the width of each rectangle is (9 - 5)/2 = 2.
5. Calculate the area of each rectangle by multiplying the height by the width. For the first rectangle, the area is (2/6) * 2, and for the second rectangle, the area is (2/8) * 2.
6. Add up the areas of both rectangles to get the total estimated area.
Now, let's repeat the process with four rectangles:
1. Divide the interval [5, 9] into four equal subintervals: [5, 6], [6, 7], [7, 8], [8, 9].
2. Find the midpoint of each subinterval. The midpoints are 5.5, 6.5, 7.5, and 8.5, respectively.
3. Calculate the height of each rectangle by evaluating the function at the corresponding midpoints. For example, the height for the first rectangle is f(5.5) = 2/5.5, and so on.
4. Calculate the width of each rectangle. Since we have divided the interval into four equal subintervals, the width of each rectangle is (9 - 5)/4 = 1.
5. Calculate the area of each rectangle by multiplying the height by the width. For example, the area of the first rectangle is (2/5.5) * 1, and so on.
6. Add up the areas of all four rectangles to get the total estimated area.
By using the midpoint rule with two and four rectangles, you can estimate the area under the graph of the function f(x) = 2/x between x = 5 and x = 9.