/*! 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 1 Find the volume of the solid wit... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 1

Find the volume of the solid with cross-sectional area \(A(x)\). $$A(x)=x+2,-1 \leq x \leq 3$$

Short Answer

Expert verified
The volume of the solid with the cross-sectional area \(A(x) = x + 2\) over the interval \(-1 \leq x \leq 3\) is 12 cubic units.

Step by step solution

01

Identify the Area Function and the Interval

The area function provided is \(A(x) = x + 2\) and the interval for which we have to find the solid volume is \(-1 \leq x \leq 3\) .
02

Setup the Integral

Set up the integral using the area function and the interval. This is done because the integral of an area function over a specified interval gives the volume of the solid. The setup looks like this: \(\int_{-1}^{3} (x+2) dx\) .
03

Apply the Fundamental Theorem of Calculus

Use the Fundamental Theorem of Calculus to evaluate the integral. First, find the antiderivative (F(x)) of the integrand (x+2). This gives \(F(x) = 0.5x^2 + 2x\). Then evaluate F(x) at the upper and lower limits of the integral and subtract the two results. This looks like this: \(F(3) - F(-1) = [0.5(3)^2 + 2(3)] - [0.5(-1)^2 + 2(-1)]\)
04

Evaluate and Simplify

Now, evaluate and simplify the expression for the volume. Carry out the arithmetic operations to find the numeric answer: \(4.5 + 6 - 0.5 + 2 = 12\) .

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.

Cross-Sectional Area
A cross-sectional area represents a two-dimensional slice through a three-dimensional object. It's like slicing a loaf of bread and looking at the shape of the slices. In calculus, this concept helps in understanding how solids have been physically structured from these areas. The exercise uses the formula for cross-sectional area, given as \(A(x) = x + 2\).
This means at any point \(x\) in the interval from \(-1\) to \(3\), if you cut through the solid, the area of that cut is \(x + 2\).
  • The formula shows how the area changes as \(x\) changes.
  • The expression \(x + 2\) increases linearly, showing a direct but simple variation with \(x\).
Knowing this function helps in determining the volume of a solid by integrating the cross-sectional area function over an interval.
Definite Integral
A definite integral is a powerful tool in calculus that adds up small quantities to find a total amount, such as an area under a curve. In the context of finding volume, it helps to accumulate the infinite tiny areas of cross-sections over a certain interval. To find the volume of a solid from its cross-sectional area, the integral is set up as \(\int_{-1}^{3} (x+2) dx\).
  • The numbers \(-1\) and \(3\) are the limits of integration, defining where to start and stop on the \(x\)-axis.
  • The expression \((x+2) dx\) represents the tiny cross-sectional area that is being added up across the interval.
Through the definite integral, we convert the changing cross-sectional areas into a total volume.
It's like summing up all the slices of a loaf to get the full loaf.
Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus forms the bridge between differentiation and integration. It provides a way to evaluate a definite integral using antiderivatives.First, find the antiderivative of the function inside the integral, such as for \(x + 2\):
  • The antiderivative is \(F(x) = 0.5x^2 + 2x\).
  • It represents a function whose derivative gives back the integrand \(x+2\).
Once this function is established, calculate it at the integral's upper limit \(3\) and lower limit \(-1\),
  • Then subtract: \(F(3) - F(-1)\).
  • This solution gives the exact volume, by effectively summing the total accumulation of the cross-sectional areas from \(-1\) to \(3\).
The process highlights how integration, coupled with antiderivatives, can provide concrete sums for continuous, changing quantities.

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

Suppose that the circle \(x^{2}+y^{2}=1\) is revolved about the y-axis. Show that the volume of the resulting solid is \(\frac{4}{3} \pi\)

Find the indicated probabilities, given that the lifetime of a light bulb is exponentially distributed with pdf \(f(x)=6 e^{-6 x}\) (with \(x\) measured in years). The light bulb lasts less than 3 months.

The base of solid \(V\) is the region bounded by \(y=x^{2}\) and \(y=\sqrt{x} .\) Find the volume if \(V\) has (a) square cross sections and (b) semicircular cross sections perpendicular to the \(x\) -axis.

If the probability of an event is \(p\), the probability that it will happen \(m\) times in \(n\) tries is \(f(p)=\frac{n !}{m !(n-m) !} p^{m}(1-p)^{n-m}\) Find the value of \(p\) that maximizes \(f(p) .\) This is called the maximum likelihood estimator of \(p .\) Briefly explain why your answer makes sense.

Set up the integral for the surface area of the surface of revolution and approximate the integral with a numerical method. \(y=\sqrt{x}, 1 \leq x \leq 2,\) revolved about the \(x\) -axis
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

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.