Suggested languages for you:

Americas

Europe

Q2E

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