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Chapter 7: Problem 24

In Exercises 23-28, find the volume of the solid generated by revolving the region about the y-axis. the region enclosed by $$x=y^{3 / 2}, x=0, y=2$$

Short Answer

Expert verified
The volume of the solid is \(4\pi\) cubic units.

Step by step solution

01

Set Up the Integral

The integral will be from 0 to 2 because these are the limits of y. We will square the equation \(x = y^{3 / 2}\) and multiply by \(\pi\) to find the volume. So, the integral is \(\int_{0}^{2} \pi * (y^{3 / 2})^2 dy\).
02

Simplify the Integral

The integral simplifies to \(\int_{0}^{2} \pi * y^3 dy\).
03

Evaluate the Integral

The antiderivative of \(y^3\) is \(\frac{1}{4}y^4\). So, the integral becomes \(\frac{1}{4} \pi * y^4 |_{0}^{2}\)
04

Calculate the Result

The final result of the integral is \(\frac{1}{4} \pi * 2^4 - \frac{1}{4} \pi * 0^4 = \frac{1}{4} \pi * 16 = 4\pi\).

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Key Concepts

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

Integral Calculus
Integral calculus is a branch of mathematics focused on functions and the accumulation of quantities. In this problem, we use it to calculate the volume of a solid through a process called the volume of revolution. We integrate functions to find areas, lengths, and volumes.
When dealing with volumes of revolution, we create a solid by rotating a curve around an axis. The integral helps us sum infinitely many slices of these solids to find the precise volume.
  • We start by defining the limits of integration, which indicate the interval over which the function is revolved.
  • Next, we square the function and multiply by \( \pi \) to represent the stacked disks forming the solid.
This method allows us to easily compute complex 3D shapes using simple 2D functions.
Solid of Revolution
A solid of revolution is what you get when you take a 2D area and spin it around an axis, forming a 3D object. In this exercise, we're revolving the curve \( x = y^{3/2} \) around the y-axis.
To visualize, picture a curve on a graph. When revolved, this curve sweeps out a complete shape around the axis. The region contained within this rotation forms the solid.
  • Imagine revolving the line \( x = y^{3/2} \) from \( x = 0 \) to \( y = 2 \). It creates a hollow, vase-like shape as it spins around the y-axis.
  • The integral captures this 3D rotation by summing up slices of cylindrical disks in the shape's structure.
This concept helps in visualizing volume calculation, turning abstract math into something tangible.
Antiderivative
The antiderivative, also known as the indefinite integral, is used to reverse the process of differentiation. In our solution, calculating the antiderivative of a function is crucial to finding the volume.
When we look for the antiderivative of \( y^3 \), we seek a function whose derivative returns \( y^3 \) itself. In this case, the antiderivative of \( y^3 \) is \( \frac{1}{4}y^4 \).
  • The antiderivative allows us to evaluate definite integrals, giving a concrete value to the problem.
  • By plugging in the upper and lower limits after finding the antiderivative, we determine the entire volume from start (\( y = 0 \)) to finish (\( y = 2 \)).
Thus, the antiderivative plays a key role in transforming a theoretical setup into real-world measurements.

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Most popular questions from this chapter

True or False The volume of a solid of a known integrable cross section area \(A(x)\) from \(x=a\) to \(x=b\) is \(\int_{a}^{b} A(x) d x .\) Justify your answer.

Find the area of the propeller-shaped region enclosed between the graphs of $$y=\sin x$$ and $$y=x^{3}$$ .

Kinetic Energy If a variable force of magnitude \(F(x)\) moves a body of mass \(m\) along the \(x\) -axis from \(x_{1}\) to \(x_{2},\) the body's velocity \(v\) can be written as \(d x / d t\) (where \(t\) represents time). Use Newton's second law of motion, \(F=m(d v / d t),\) and the Chain Rule $$\frac{d v}{d t}=\frac{d v}{d x} \frac{d x}{d t}=v \frac{d v}{d x}$$ to show that the net work done by the force in moving the body from \(x_{1}\) to \(x_{2}\) is $$W=\int_{x_{1}}^{x_{2}} F(x) d x=\frac{1}{2} m v_{2}^{2}-\frac{1}{2} m v_{1}^{2}$$ where \(v_{1}\) and \(v_{2}\) are the body's velocities at \(x_{1}\) and \(x_{2} .\) In physics the expression \((1 / 2) m v^{2}\) is the kinetic energy of the body moving with velocity v. Therefore, the work done by the force equals the change in the body's kinetic energy, and we can find the work by calculating this change. Weight vs. Mass Weight is the force that results from gravity pulling on a mass. The two are related by the equation in Newton's second law, weight \(=\) mass \(\times\) acceleration. Thus, newtons \(=\) kilograms \(\times \mathrm{m} / \mathrm{sec}^{2}\) , pounds \(=\) slugs \(\times \mathrm{ft} / \mathrm{sec}^{2}\) . To convert mass to weight, multiply by the acceleration of gravity. To convert weight to mass, divide by the acceleration of gravity.

The Classical Bead Problem A round hole is drilled through the center of a spherical solid of radius \(r .\) The resulting cylindrical hole has height 4 \(\mathrm{cm} .\) (a) What is the volume of the solid that remains? (b) What is unusual about the answer?

Writing to Learn Suppose that \(f(t)\) is the probability density function for the lifetime of a certain type of lightbulb where \(t\) is in hours. What is the meaning of the integral $$\int_{100}^{800} f(t) d t ?$$
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