/*! 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 13 Evaluate \(\int_{1}^{4}\left(\fr... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 31: Problem 13

Evaluate \(\int_{1}^{4}\left(\frac{\theta+2}{\sqrt{\theta}}\right) \mathrm{d} \theta\), taking positive square roots only.

Short Answer

Expert verified
The value of the integral is \( \frac{20}{3} \).

Step by step solution

01

Simplify the Integrand

We start by simplifying the given integrand. The function to integrate is \( \frac{\theta + 2}{\sqrt{\theta}} \). We can split this into two separate terms: \( \frac{\theta}{\sqrt{\theta}} + \frac{2}{\sqrt{\theta}} \). Simplifying each term gives \( \theta^{1/2} + 2\theta^{-1/2} \).
02

Set Up the Integral

With the expression simplified, set up the integral for each term separately: \[ \int_{1}^{4} \theta^{1/2} \,d\theta + 2 \int_{1}^{4} \theta^{-1/2} \,d\theta \]
03

Integrate Each Term

Integrate each term using the power rule for integrals:- For \( \int \theta^{1/2} \,d\theta \), the result is \( \frac{2}{3} \theta^{3/2} \).- For \( \int \theta^{-1/2} \,d\theta \), the result is \( 2\theta^{1/2} \).
04

Substitute Boundaries and Evaluate

For each integrated term, substitute the upper and lower bounds of the integral (4 and 1) and evaluate:- For \( \frac{2}{3} \theta^{3/2} \) from 1 to 4, compute: \( \frac{2}{3} \times 4^{3/2} - \frac{2}{3} \times 1^{3/2} = \frac{2}{3} \times 8 - \frac{2}{3} \times 1 = \frac{16}{3} - \frac{2}{3} = \frac{14}{3} \).- For \( 2\theta^{1/2} \) from 1 to 4, compute: \( 2 \times 4^{1/2} - 2 \times 1^{1/2} = 2 \times 2 - 2 = 4 - 2 = 2 \).
05

Sum the Results

Sum the results from each evaluated integration:\[ \frac{14}{3} + 2 = \frac{14}{3} + \frac{6}{3} = \frac{20}{3} \].

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.

Integrand Simplification
Integrand simplification is a core step in making complex integrals easier to solve. In this exercise, the integrand given is \( \frac{\theta + 2}{\sqrt{\theta}} \), which at first glance, looks a bit tricky. However, by breaking it down, the expression becomes much simpler to handle. To simplify, divide the numerator terms individually by the denominator. This results in:
  • \( \frac{\theta}{\sqrt{\theta}} \) which simplifies to \( \theta^{1/2} \).
  • \( \frac{2}{\sqrt{\theta}} \) which simplifies to \( 2\theta^{-1/2} \).
Simplifying the integrand into separate terms allows us to apply known integration techniques more efficiently. The new, simplified integrand helps in setting up the integral correctly for further steps.
Power Rule for Integration
The power rule for integration is a fundamental tool in calculus. It simplifies the process of finding the integral of functions that are powers of a variable. The general power rule for integration states: Given a function \( f(x) = x^n \), the integral is \( \int x^n \,dx = \frac{1}{n+1} x^{n+1} + C \), where \( n eq -1 \).Applying this rule, we can integrate each part of the simplified integrand:
  • For \( \theta^{1/2} \), the integral \( \int \theta^{1/2} \,d\theta \) becomes \( \frac{2}{3} \theta^{3/2} \).
  • For \( 2\theta^{-1/2} \), the integral \( 2 \int \theta^{-1/2} \,d\theta \) becomes \( 2 \times 2\theta^{1/2} = 4\theta^{1/2} \).
The power rule transforms these terms into functions that are manageable for evaluating definite integrals.
Evaluating Definite Integrals
Evaluating a definite integral means finding the net area under a curve between two specified bounds. This involves substituting the limits into the integrated function and calculating their difference.For our problem, the limits are 1 and 4. We proceed as follows:
  • For \( \frac{2}{3} \theta^{3/2} \), substitute \( \theta = 4 \) and \( \theta = 1 \) to calculate \( \left( \frac{2}{3} \times 4^{3/2} \right) - \left( \frac{2}{3} \times 1^{3/2} \right) = \frac{14}{3} \).
  • For \( 4\theta^{1/2} \), substitute to find \( \left( 4 \times 4^{1/2} \right) - \left( 4 \times 1^{1/2} \right) = 2 \).
After evaluating both integrals, sum the results to find the total area. In this case, the final result is the sum \( \frac{14}{3} + 2 = \frac{20}{3} \). This process shows how definite integrals provide a precise calculation of area within specified boundaries.

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

Determine $$ \int\left(4+\frac{3}{7} x-6 x^{2}\right) \mathrm{d} x $$

Determine (a) \(\int 4 \cos 3 x \mathrm{~d} x\) (b) \(\int 5 \sin 2 \theta \mathrm{d} \theta\)

Determine \(\int 3 \sqrt{x} \mathrm{~d} x\)

Determine \(\int \frac{(1+\theta)^{2}}{\sqrt{\theta}} \mathrm{d} \theta\)

Evaluate (a) \(\int_{1}^{2} 4 \mathrm{e}^{2 x} \mathrm{~d} x\) (b) \(\int_{1}^{4} \frac{3}{4 u} \mathrm{~d} u\), each correct to 4 significant figures.
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Discrete 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.