Consider the given function. f(x) = 3 - (1/2)x. Evaluate the Riemann sum for 2 less than or equal...

Question:

Consider the given function.

{eq}f(x) = 3 - \frac{1}{2}x {/eq}

Evaluate the Riemann sum for {eq}2 \leq x \leq 14 {/eq}, with six subintervals, taking the sample points to be left endpoints.

Rieman Sum:

Given a function {eq}f(x) {/eq}, we approximate its integral over the interval {eq}[x_0,x_1] {/eq} as the sum of the area of rectangles.

To this end, the interval is partitioned in contiguous and non overlapping intervals {eq}[x_{n-1}, x_n] {/eq}

The integral is approximated by the Rieman sum with left endpoints as sample points

{eq}A_{left} = \sum_{n} f(x_{n-1})\Delta {/eq}

where {eq}\Delta {/eq} is the sample spacing.

For the problem at hand, we have

{eq}x_0 =2 \\ x_1 = 14 \\ f(x) = 3 - (1/2)x {/eq}

The interval between {eq}[2,14] {/eq} is discretized into six intervals of equal width {eq}\Delta = 2 {/eq}

The integral approximated with left Rieman sums is

{eq}A = \sum_{n} f(x_{n-1})\Delta = 2(f(2) + f(4) + f(6) + f(8) + f(10) + f(12)) = -6 {/eq}