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Chapter 5: Problem 1

Find the volume generated by rotating the area bounded by the graphs of each set of equations around the \(x\) -axis. $$ y=x, x=0, x=1 $$

Short Answer

Expert verified
The volume is \( \frac{\pi}{3} \).

Step by step solution

01

Understand the Region

We need to visualize the region bounded by the equations in the xy-plane. The line \( y = x \) from \( x = 0 \) to \( x = 1 \) is a straight line forming a right triangle with the x-axis. The base runs from \( x = 0 \) to \( x = 1 \) on the x-axis, and the height of the triangle is the same, reaching up to \( y = 1 \).
02

Set Up the Volume Integral

To find the volume of the solid generated by rotating the region around the x-axis, we use the disk method. The formula for the volume \( V \) of a solid of revolution about the x-axis is: \[ V = \pi \int_{a}^{b} [f(x)]^2 \, dx \] Here, \( f(x) = x \), \( a = 0 \), and \( b = 1 \).
03

Integrate to Find the Volume

Substitute the function and limits into the integral. \[ V = \pi \int_{0}^{1} (x)^2 \, dx \] To evaluate the integral, compute \int x^2 \, dx: \[ \int x^2 \, dx = \frac{x^3}{3} \] Evaluate this from 0 to 1: \[ V = \pi \left[ \frac{x^3}{3} \right]_{0}^{1} = \pi \left( \frac{1^3}{3} - \frac{0^3}{3} \right) \] \[ V = \frac{\pi}{3} \]
04

State the Volume

The volume of the solid generated by rotating the region bounded by \( y = x \), \( x = 0 \), and \( x = 1 \) around the x-axis is \( \frac{\pi}{3} \).

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Key Concepts

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

Disk Method
The Disk Method is a technique used to find the volume of a solid of revolution. This method is particularly useful when you are revolving a region around one of the coordinate axes. Imagine breaking up that region into many thin slices, shaped like disks. Each disk has a small thickness, typically denoted by \( dx \). The idea is to sum up the volumes of all these disks to determine the entire volume of the solid.To apply the Disk Method:
  • Visualize the region to be revolved. This helps you see how each slice contributes to the solid as a whole.
  • Find the radius of each disk, which is determined by the function that describes the boundary of your region. In this situation, the function is \( f(x) = x \).
  • Use the formula for the volume of a disk, which uses the area formula \( \pi [f(x)]^2 \). Multiply it by the thickness, \( dx \), and integrate from the lower to the upper boundary of your region.
This results in the formula: \[ V = \pi \int_{a}^{b} [f(x)]^2 \, dx \]This integrates the volumes of all the disks to give the whole volume of the solid of revolution.
Definite Integral
The definite integral is a powerful mathematical tool used to compute the net area under a curve over a specific interval, \([a, b]\). When applied to the Disk Method, it helps in calculating the total volume of a solid. The bounds \(a\) and \(b\) describe the limits of integration, which denote where the accumulation starts and ends along the x-axis.Here's how a definite integral works in this context:
  • The integral symbol \( \int \) represents the summation process. It adds up infinitely small quantities from the starting point \(a\) to the endpoint \(b\), across the interval.
  • The integrand, which is \((x)^2\) in this situation, describes the function whose area or, in this instance, volume is being calculated.
  • To evaluate, apply the rules of integration to find an antiderivative (a function whose derivative results in the original integrand) and calculate it at the limits \(a\) and \(b\).
For example, the integral \[ \int_{0}^{1} x^2 \, dx = \frac{x^3}{3} \] when evaluated from 0 to 1 results in \( \frac{1}{3} - 0 \), revealing the total volume contribution of the specified interval.
Function Rotation
Function rotation involves creating a three-dimensional solid by revolving a two-dimensional region around an axis. This transformation is what allows us to apply concepts like the Disk Method to find the volume of these solids.Key aspects of Function Rotation include:
  • Choosing the Axis: Decide whether the axis of rotation is horizontal or vertical, which influences the choice of integration method and setup.
  • Determining the Radius: The radius of rotation is the distance from the axis to the function or to the edge of the region being revolved. In this problem, it's simply the value of \(x\), the function itself.
  • Visualizing the Solid: Understanding the shape and boundaries of the resulting three-dimensional object helps in setting up the correct calculus problem and ensuring the accuracy of integration limits.
By rotating a function like \(y = x\) around the x-axis from \(x = 0\) to \(x = 1\), you form a solid that resembles a cone or trumpet-like shape. Calculating its volume helps in understanding how areas in two dimensions translate into spaces in three dimensions.

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Most popular questions from this chapter

The validity of the WeberFechner Law has been the subject of great debate among psychologists. An alternative model, $$ \frac{d R}{d S}=k \cdot \frac{R}{S} $$ where \(k\) is a positive constant, has been proposed. Find the general solution of this equation. (This model has also been referred to as the Power Law of Stimulus-Response.)

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?

Look up some data on rate of use and current world reserves of a natural resource not considered in this section. Predict when the world reserves for that resource will be depleted.

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. $$ f(x)=\frac{1}{4} x, \quad[1,3] $$

Suppose 30 sparrows are released into a region where they have no natural predators. The growth of the region's sparrow population can be modeled by the uninhibited growth model \(d P / d t=k P\) where \(P(t)\) is the population of sparrows \(t\) years after their initial release. a) When the sparrow population is estimated at \(12,500,\) its rate of growth is about 1325 sparrows per year. Use this information to find \(k,\) and then find the particular solution of the differential equation. b) Find the number of sparrows after 70 yr. c) Without using a calculator, find \(P^{\prime}(70) / P(70)\).
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