/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 50 Find \(\int_{0}^{5} f(x) d x\) i... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 50

Find \(\int_{0}^{5} f(x) d x\) if $$ f(x)=\left\\{\begin{array}{ll}{3} & {\text { for } x<3} \\ {x} & {\text { for } x \geqslant 3}\end{array}\right. $$

Short Answer

Expert verified
The value of the integral is 17.

Step by step solution

01

Understand the Piecewise Function

The function \( f(x) \) is defined as a piecewise function. For values of \( x \) from 0 to less than 3, \( f(x) = 3 \). For values of \( x \) greater than or equal to 3, \( f(x) = x \).
02

Split the Integral

Since the function is defined piecewise, split the integral at the point where the definition of the function changes, from 0 to 3 and from 3 to 5. The integral can be written as: \[ \int_{0}^{5} f(x) \, dx = \int_{0}^{3} 3 \, dx + \int_{3}^{5} x \, dx. \]
03

Integrate the First Interval

Calculate the integral of the constant function over the first interval: \[ \int_{0}^{3} 3 \, dx = 3 \times (3 - 0) = 9. \]
04

Integrate the Second Interval

Calculate the integral over the second interval: \[ \int_{3}^{5} x \, dx = \left[ \frac{x^2}{2} \right]_{3}^{5}. \] Evaluate the integral: \[ \frac{5^2}{2} - \frac{3^2}{2} = \frac{25}{2} - \frac{9}{2} = \frac{16}{2} = 8. \]
05

Sum the Results

Add the results of the two integrals from Step 3 and Step 4: \[ 9 + 8 = 17. \]

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.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Piecewise Functions
A piecewise function is a function composed of different expressions over different intervals. These are primarily useful when a function has distinct behaviors over its domain. In our problem, the piecewise function is defined as follows:
  • For values of \( x \) less than 3, \( f(x) = 3 \). This means from 0 to less than 3, the function is a constant horizontal line at \( y = 3 \).
  • For values of \( x \) that are 3 or greater, \( f(x) = x \). Here, the function is simply the line \( y = x \).
The goal when working with piecewise functions is to ensure we integrate each piece over its respective interval before summing the results. This approach is essential because it considers the different expressions the function takes.
The Concept of Definite Integrals
Definite integrals are an important concept in calculus, allowing us to find the net area under a curve from one point to another along the x-axis. Unlike indefinite integrals, which include a constant of integration, definite integrals yield a numeric result that represents the total accumulated value over a specified interval.When working with a piecewise function, as seen in our exercise, the definite integral must be split into separate integrals over each defined segment:
  • From 0 to 3 for the constant function \( f(x) = 3 \).
  • From 3 to 5 for the linear function \( f(x) = x \).
By evaluating these separate definite integrals and adding them together, we accurately determine the area under the piecewise curve from 0 to 5.
Basics of Calculus and Integration
Calculus is a branch of mathematics that deals with change and motion, utilizing concepts such as derivatives and integrals. One of the core purposes of calculus is to understand how things accumulate; integration specifically helps calculate the total value of a function over an interval.In our exercise, integration is used to accumulate the total value of the piecewise function over its domain. Here, integration encompasses:
  • Calculating the area under the constant segment of the function, which results in simply multiplying the value by the width of the interval.
  • Applying the power rule for integration to determine the accumulated value of the linear segment \( f(x) = x \).
Through proper integration techniques, calculus allows us to dissect complex problems into manageable parts, find solutions, and understand deeper relationships within mathematics.

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 \(f(x)\) is the slope of a trail at a distance of \(x\) miles from the start of the trail, what does \(\int_{3}^{5} f(x) d x\) represent?

Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative (take \(C=0) .\) $$ \int e^{\cos x} \sin x d x $$

Evaluate the indefinite integral. $$ \int \frac{d x}{a x+b}(a \neq 0) $$

If \(f(x)\) is the slope of a trail at a distance of \(x\) miles from the start of the trail, what does \(\int_{3}^{5} f(x) d x\) represent?

Evaluate the integral. $$ \int_{0}^{\pi / 3} \frac{\sin \theta+\sin \theta \tan ^{2} \theta}{\sec ^{2} \theta} d \theta $$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

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.