Let f (x) = 5 - x/2. What is the value of the Riemann sum R(f ,0,8), obtained by using four rectangles and left-hand endpoints?
To calculate the Riemann sum using left-hand endpoints, we divide the interval [0, 8] into four equal subintervals. Each rectangle will have a width of 8/4 = 2.
The left-hand endpoints for the four subintervals are 0, 2, 4, and 6.
To find the height of each rectangle, we evaluate the function f(x) = 5 - x/2 at each of the left-hand endpoints.
- For x = 0, f(0) = 5 - 0/2 = 5.
- For x = 2, f(2) = 5 - 2/2 = 4.
- For x = 4, f(4) = 5 - 4/2 = 3.
- For x = 6, f(6) = 5 - 6/2 = 2.
So, the heights of the four rectangles are 5, 4, 3, and 2.
Now, we can calculate the Riemann sum by multiplying the width and height of each rectangle and then summing those values:
R(f, 0, 8) = (2 * 5) + (2 * 4) + (2 * 3) + (2 * 2)
= 10 + 8 + 6 + 4
= 28.
Therefore, the value of the Riemann sum R(f, 0, 8) obtained by using four rectangles and left-hand endpoints is 28.