Question

If f (x) = √x - 2, 1 ≤  x ≤ 6, find the Riemann sum with n = 5 correct to six decimal places, taking the sample points to be midpoints. What does the Riemann sum represent? Illustrate with a diagram.

Answer 1

Step 1 :

Definition :

If f is integrable on , then

Where and .

Step 2 :

The function is and the interval is .

Find the interval width with and .

Subintervals are and .

Midpoints are and .

Find the Riemann sum :

Step 3 :

Graph the function :

 

The Riemann sum represents the sum of the areas of the two rectangles above the x - axis minus the sum of the areas of the three rectangles below the x - axis.

Riemann sum is the net area of the rectangles with respect to the x - axis.

Solution :