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Chapter 5: Problem 13

Find the volume generated by rotating the area bounded by the graphs of each set of equations around the \(x\) -axis. $$ y=x^{2}, x=0, x=2 $$

Short Answer

Expert verified
The volume is \( \frac{32\pi}{5} \).

Step by step solution

01

Understand the Problem

We need to find the volume of the solid formed by rotating the region bounded by the curves \( y = x^2 \), \( x = 0 \), and \( x = 2 \) around the \( x \)-axis.
02

Set Up the Integral

The formula for the volume of a solid of revolution around the \( x \)-axis is \( V = \pi \int_{a}^{b} [f(x)]^2 \, dx \). Here \( f(x) = x^2 \), \( a = 0 \), and \( b = 2 \), so the integral becomes \( V = \pi \int_{0}^{2} (x^2)^2 \, dx = \pi \int_{0}^{2} x^4 \, dx \).
03

Compute the Integral

Calculate the integral \( \int_{0}^{2} x^4 \, dx \). Find the antiderivative: \( \int x^4 \, dx = \frac{x^5}{5} + C \). Now evaluate from 0 to 2: \[ \left. \frac{x^5}{5} \right|_0^2 = \frac{2^5}{5} - \frac{0^5}{5} = \frac{32}{5} \].
04

Calculate the Volume

Substitute the result of the integral back into the formula: \( V = \pi \times \frac{32}{5} = \frac{32\pi}{5} \). This is the volume of the solid obtained by rotating the region around the \( x \)-axis.

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Key Concepts

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

Volume of Revolution
When we talk about finding the volume of a solid of revolution, we're referring to a process where a 2D region is spun around an axis to create a 3D object. This is a common technique in calculus to measure the volume of irregular shapes. Let's break it down a bit:
  • First, identify the region to be rotated. In this case, the region is bounded by the graph of the function, such as \( y = x^2 \), within certain limits (here, \( x=0 \) to \( x=2 \)).
  • This region is rotated about an axis (the \( x \)-axis in our example), sweeping out a volume.
  • The shape formed by this rotation is similar to a series of infinitely thin disks or washers stacked together.
To determine this volume precisely, we apply calculus, as the simple multiplication formulas for geometric shapes don't directly apply here. Hence, the method called the disk or washer method uses integration to handle this.
Definite Integral
The definite integral is a key tool used in calculus to compute areas, volumes, and more. Essentially, the definite integral calculates the accumulated sum of quantities, such as area under a curve or in our case, volume of a solid of revolution. Here鈥檚 a simplified breakdown:
  • Definition: A definite integral from \(a\) to \(b\) is represented as \( \int_{a}^{b} f(x) \, dx \) and measures the net area under the curve of \(f(x)\) from \(x = a\) to \(x = b\).
  • In our example, the integral \( \pi \int_{0}^{2} (x^2)^2 \, dx \) calculates the volume sweeping out around the \(x\)-axis, transforming our 2D shape into a 3D solid.
  • The result we obtain, through careful integration and accounting for the limits, directly gives a concrete volume of the space revolutionized by the function about the axis.
Integrating, in essence, sums up infinitely small quantities to provide an exact and complete measurement within the defined limits.
Antiderivative
An antiderivative, also known as an indefinite integral, is essentially the reverse process of differentiation. It's a function whose derivative is the original function you're considering. Understanding antiderivatives is crucial for solving integration problems:
  • In our problem, finding the volume required us to first determine the antiderivative of \( x^4 \), which represents the solid's cross-sectional area when expressed as \( [f(x)]^2 \).
  • The process involved finding a function whose derivative returns \( x^4 \). For polynomials, a function like \( \int x^4 \, dx = \frac{x^5}{5} + C \) does this, where \(C\) is a constant of integration that is determined by evaluating the definite endpoints.
Once the antiderivative is determined, it鈥檚 evaluated at the upper and lower limits of the integral to calculate the total area under the curve or, as in this exercise, to find the total solid volume. This process showcases how intricately calculus techniques are woven into finding practical, physical quantities.

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Most popular questions from this chapter

(a) write a differential equation that models the situation, and (b) find the general solution. If an initial condition is given, find the particular solution. Recall that when \(y\) is directly proportional to \(x,\) we have \(y=k x\), and when \(y\) is inversely proportional to \(x,\) we have \(y=k / x,\) where \(k\) is the constant of proportionality. In these exercises, let \(k=1\). The rate of change of \(y\) with respect to \(x\) is directly proportional to the cube of \(y\).

Jennifer deposits \(A_{0}=\$ 1200\) into an account that earns \(4.2 \%\) compounded continuously. a) Write the differential equation that represents \(A(t)\), the value of Jennifer's account after t years. b) Find the particular solution of the differential equation from part (a). c) Find \(A(7)\) and \(A^{\prime}(7)\). d) Find \(A^{\prime}(7) / A(7)\), and explain what this number represents.

For each probability density function, over the given interval, find \(\mathrm{E}(x), \mathrm{E}\left(x^{2}\right),\) the mean, the variance, and the standard deviation. $$ f(x)=\frac{1}{8} x, \quad[0,4] $$

Gusto Stick is a professional baseball player who has just become a free agent. His attorney begins negotiations with an interested team by asking for a contract that provides Gusto with an income stream given by \(R_{1}(t)=800,000+340,000 t,\) over \(10 \mathrm{yr}\), where \(t\) is in years. (Round all answers to the nearest \(\$ 100 .)\) a) What is the accumulated future value of the offer. assuming an interest rate of \(5 \%,\) compounded continuously? b) What is the accumulated present value of the offer, assuming an interest rate of \(5 \%,\) compounded continuously? c) The team counters by offering an income stream given by \(R_{2}(t)=600,000+210,000 t\). What is the accumulated present value of this counteroffer? d) Gusto comes back with a demand for an income stream given by \(R_{3}(t)=1,000,000+250,000 t\). What is the accumulated present value of this income stream? e) Gusto signs a contract for the income stream in part (d) but decides to live on \(\$ 500,000\) each year, investing the rest at \(5 \%,\) compounded continuously. What is the accumulated future value of the remaining income, assuming an interest rate of \(5 \%,\) compounded continuously?

The amount of money, \(A(t)\), in Ina's savings account after \(t\) years is modeled by the differential equation \(d A / d t=0.0418 A\) a) What is the continuous growth rate? b) Find the particular solution, \(A(t),\) if Ina's account is worth \(\$ 3479.02\) after 2 yr. c) Find the amount that Ina deposited initially.
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