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Expert-verified Found in: Page 452 ### Essential Calculus: Early Transcendentals

Book edition 2nd
Author(s) James Stewart
Pages 830 pages
ISBN 9781133112280 # Suppose is a continuous positive decreasing function for $$x \ge 1$$ and . By drawing a picture, rank the following three quantities in increasing order:$$\int\limits_1^6 {f(x)dx}$$ $$\sum\limits_{i = 1}^5 {\mathop a\nolimits_i }$$ $$\sum\limits_{i = 2}^6 {\mathop a\nolimits_i }$$

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

See the 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, ﻿$$\int\limits_1^6 {f(x)dx}$$<$$\sum\limits_{i = 1}^5 {\mathop a\nolimits_i }$$<$$\sum\limits_{i = 2}^6 {\mathop a\nolimits_i }$$ ### Want to see more solutions like these? 