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Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 6 - Applications of IntegrationProblem 6.R.30
Chapter 6, Problem 6.R.30

27–33. Multiple regions The regions R₁,R₂, and R₃ (see figure) are formed by the graphs of y = 2√x,y = 3−x,and x=3.


Find the volume of the solid obtained by revolving region R₂ about the y-axis.
Graph showing regions R₁, R₂, and R₃ bounded by y=2√x, y=3−x, and x=3 on xy-plane.

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Identify the region R₂ bounded by the curves y = 2\(\sqrt{x}\), y = 3 - x, and the vertical line x = 3. This region lies between the curves y = 2\(\sqrt{x}\) (lower curve) and y = 3 - x (upper curve) from x = 1 to x = 3, where the two curves intersect.
Since the solid is formed by revolving region R₂ about the y-axis, use the method of cylindrical shells. The formula for the volume using cylindrical shells when revolving around the y-axis is: \[ V = 2\pi \int_{a}^{b} (\text{radius})(\text{height}) \, dx \] where the radius is the distance from the y-axis (which is x), and the height is the vertical distance between the two curves.
Determine the radius and height for the shell at a typical x between 1 and 3: - Radius = x - Height = (upper curve) - (lower curve) = (3 - x) - 2\(\sqrt{x}\)
Set up the integral for the volume: \[ V = 2\pi \int_{1}^{3} x \left[(3 - x) - 2\sqrt{x}\right] \, dx \]
Evaluate the integral by expanding the integrand and integrating term-by-term with respect to x over the interval [1, 3]. This will give the volume of the solid formed by revolving region R₂ about the y-axis.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Volume of Solids of Revolution

This concept involves finding the volume of a 3D solid formed by rotating a 2D region around an axis. The volume can be computed using methods like the disk/washer or shell method, depending on the axis of rotation and the shape of the region.
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Finding Volume Using Disks

Shell Method

The shell method calculates volume by integrating cylindrical shells formed by revolving vertical slices of the region around the y-axis. Each shell's volume is 2π(radius)(height)(thickness), making it ideal when rotating around the y-axis and the region is described in terms of x.
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Euler's Method

Setting up Integration Limits and Functions

Accurately determining the bounds of integration and the expressions for radius and height is crucial. For region R₂, bounded by y=3−x and y=2√x, and x=3, the limits are based on the intersection points of these curves, ensuring the integral correctly represents the volume.
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Limits of Rational Functions: Denominator = 0
Related Practice
Textbook Question

14–25. {Use of Tech} Areas of regions Determine the area of the given region.


The region bounded by y = x²,y = 2x²−4x, and y = 0

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Textbook Question

43–55. Volumes of solids Choose the general slicing method, the disk/washer method, or the shell method to answer the following questions.


The region bounded by the curves y = sec x and y=2, for 0 ≤ x ≤ π/3, is revolved about the x-axis. What is the volume of the solid that is generated? 

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Textbook Question

Comparing volumes Let R be the region bounded by y=1/x^p and the x-axis on the interval [1, a], where p>0 and a>1 (see figure). Let Vₓ and Vᵧ be the volumes of the solids generated when R is revolved about the x- and y-axes, respectively.


d. Find a general expression for Vᵧ in terms of a and p. Note that p=2 is a special case. What is Vᵧ when p=2?

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Textbook Question

Two methods The region R in the first quadrant bounded by the parabola y = 4-x² and coordinate axes is revolved about the y-axis to produce a dome-shaped solid. Find the volume of the solid in the following ways:


b. Apply the shell method and integrate with respect to x.

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Textbook Question

Position, displacement, and distance A projectile is launched vertically from the ground at t=0, and its velocity in flight (in m/s) is given by v(t)=20−10t. Find the position, displacement, and distance traveled after t seconds, for 0≤t≤4.

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Textbook Question

An area function Consider the functions y = x²/a and y = √x/a, where a>0. Find A(a), the area of the region between the curves.

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