Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 12: Problem 1005

The value of the integral \({ }^{5} \int_{-5}(\mathrm{x}-[\mathrm{x}]) \mathrm{dx}\) is \(\ldots \ldots\) (a) 0 (b) 5 (c) 10 (d) 15

Short Answer

Expert verified
The value of the integral \(\int_{-5}^5 (\mathrm{x}-[\mathrm{x}])\mathrm{dx}\) is 5, which corresponds to answer choice (b).

Step by step solution

01

Break The Integral Into Two Intervals

Since the function inside the integral behaves differently for negative and positive x values, we can break the given integral into two separate integrals, one for the negative x values and the other for the positive x values. So, the integral will look like this: \[ \int_{-5}^5 (\mathrm{x}-[\mathrm{x}])\mathrm{dx} = \int_{-5}^0 (\mathrm{x}-[\mathrm{x}])\mathrm{dx} + \int_{0}^5 (\mathrm{x}-[\mathrm{x}])\mathrm{dx} \]
02

Handle The Negative Interval

For the negative interval, the floor function will round down to the nearest integer. As a result, it will "jump up" by 1 at each integer value. In this case, the function will be equal to \(x - (x-1) = 1\) between each integer, with the exception of 0. Thus, the integral becomes: \[ \int_{-5}^0 (\mathrm{x}-[\mathrm{x}])\mathrm{dx} = \int_{-5}^{-4} 1\mathrm{dx} + \int_{-4}^{-3} 1\mathrm{dx} + \dots + \int_{-1}^{0} 1\mathrm{dx} \] Now, we can evaluate each individual integral and sum them up.
03

Evaluate The Negative Interval

All the integrals in the negative interval are of the form \( \int_a^b 1\mathrm{dx} \), which evaluates to \(b - a\). Hence, we can sum them up: \[ \int_{-5}^0 (\mathrm{x}-[\mathrm{x}])\mathrm{dx} = (-4 - (-5)) + (-3 - (-4)) + \dots + (0 - (-1)) = 1 + 1 + \dots + 1 = 5 \]
04

Handle The Positive Interval

For the positive interval, the floor function will "round down" the given number to the nearest integer, resulting in \(x - x = 0\). Hence, the integral becomes: \[ \int_{0}^5 (\mathrm{x}-[\mathrm{x}])\mathrm{dx} = \int_{0}^5 0\mathrm{dx} \]
05

Evaluate The Positive Interval

The integral in the positive interval is of the form \( \int_a^b 0\mathrm{dx} \), which evaluates to 0, as there is no area underneath a function that is always zero. \[ \int_{0}^5 (\mathrm{x}-[\mathrm{x}])\mathrm{dx} = 0 \]
06

Combine Both Intervals

Finally, we combine the results from both intervals: \[ \int_{-5}^5 (\mathrm{x}-[\mathrm{x}])\mathrm{dx} = \int_{-5}^0 (\mathrm{x}-[\mathrm{x}])\mathrm{dx} + \int_{0}^5 (\mathrm{x}-[\mathrm{x}])\mathrm{dx} = 5 + 0 = 5 \] So, the value of the integral is 5, which corresponds to the answer choice (b).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

If \(a<0

The area enclosed between the curves \(y=\log _{e}(x+e)\) and the coordinate axes is...... (a) 1 (b) 4 (c) 2 (d) 3

\((\pi / 2) \int_{0} \mathrm{e}^{(\sin )-1(x)} \cdot \mathrm{e}^{(\tan )-1[\sqrt\\{1-(x) 2\\} / x]} d x\) is equal to ...... (a) \((\pi / 2)\) (b) \((\pi / 2) \mathrm{e}^{(\pi / 2)}\) (c) \((\pi / 4) \mathrm{e}^{(\pi / 2)}\) (d) \(\mathrm{e}^{(\pi / 2)}\)

The area of the region bounded by curves \(x^{2}+y^{2}=4, x=1\) \(\& x=\sqrt{3}\) is..... (a) \((\pi / 3)\) sq. unit (b) \((2 \pi / 3)\) sq. unit (c) \((5 \pi / 6)\) sq. unit (d) \((4 \pi / 3)\) sq. unit

The value of the integral \(\left.\left.{ }^{1} \int_{0} \log [\sqrt{(} 1-\mathrm{x})+\sqrt{(} 1+\mathrm{x}\right)\right] \mathrm{d} \mathrm{x}\) is \(\ldots \ldots\) (a) \((1 / 2)[\log (2)-(1 / 2)+(\pi / 4)]\) (b) \((1 / 2)[\log 2-1+(\pi / 2)]\) (c) \((1 / 3)[\log 4-1+(\pi / 4)]\) (d) \((1 / 4)[\log 3-1+(\pi / 2)]\)
See all solutions

Recommended explanations on English Textbooks

History of English Language

Read Explanation

Global English

Read Explanation

Summary Text

Read Explanation

Multiple Choice Questions

Read Explanation

Research and Composition

Read Explanation

Cues and Conventions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free