Question

Suppose is a continuous positive decreasing function for\(x \ge 1\) and \(\). By drawing a picture, rank the following three quantities in increasing order:

\(\int\limits_1^6 {f(x)dx} \) \(\sum\limits_{i = 1}^5 {\mathop a\nolimits_i } \) \(\sum\limits_{i = 2}^6 {\mathop a\nolimits_i } \)

Short answer

\(\int\limits_1^6 {f(x)dx} \)<\(\sum\limits_{i = 1}^5 {\mathop a\nolimits_i } \)<\(\sum\limits_{i = 2}^6 {\mathop a\nolimits_i } \)

Step-by-step solution

Expiation:

Given: If we assume that is a continuous positive decreasing function for \(x \ge 1\) and \(\mathop a\nolimits_n = f(n)\)



To find: By drawing a picture, we will need to rank these equations in increasing order.

\(\int\limits_1^6 {f(x)dx} \) \(\sum\limits_{i = 1}^5 {\mathop a\nolimits_i } \) \(\sum\limits_{i = 2}^6 {\mathop a\nolimits_i } \)

We will draw a graph from the given equation and then solve it.

with the help of graph

With the help of graph, the three quantities are shown in picture

\(\)

Below the blue curve, the orange portion is\(\int\limits_1^6 {f(x)dx} \)and is represented by the red rectangle.\(\sum\limits_{i = 2}^6 {\mathop a\nolimits_i } \)is represented by dark orange.

Therefore, the smallest region is\(\sum\limits_{i = 2}^6 {\mathop a\nolimits_i } \)and the largest region are\(\sum\limits_{i = 1}^5 {\mathop a\nolimits_i } \)and\(\int\limits_1^6 {f(x)dx} \)

Hence, \(\int\limits_1^6 {f(x)dx} \)<\(\sum\limits_{i = 1}^5 {\mathop a\nolimits_i } \)<\(\sum\limits_{i = 2}^6 {\mathop a\nolimits_i } \)