Find the value V of the Riemann sum V = n ∑ k = 1 f (ck)Δxk for the function f ( x ) = x 2 − 1 using the partition P = { 0, 2, 5, 8 }, where ck is the left endpoints of the partition.

Answer 1

What you are asking is called the left Reimann sum since we are using the left endpoints of each interval.

The values of Δx_k are (2-0), (5-2) and (8-5). That is, 2, 3 and 3.

The values of c_k are 0, 2 and 5

The values of f(c_k) are (0²-1), (2²-1) and (5²-1). That is, -1, 3 and 24.

So, each term in the Reimann sum is f(c_k)Δx_k and they are (-1×2), (3×3) and (24×3)

These values are -2, 9 and 72.

So, adding them together, the Reimann sum is 79.