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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 6.3.26
Chapter 6, Problem 6.3.26

Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis.
y=x,y=x+2,x=0, and x=4 ; about the x-axis
Graph showing region bounded by y = x, y = x + 2, x = 0, and x = 4 shaded between two lines.

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Identify the region R bounded by the curves y = x, y = x + 2, x = 0, and x = 4. This region lies between the two lines y = x and y = x + 2, from x = 0 to x = 4.
Since the region is revolved about the x-axis, use the method of washers to find the volume. The volume element dV is given by the difference of the areas of the outer and inner disks times the thickness dx.
The outer radius R(x) is the distance from the x-axis to the upper curve y = x + 2, so R(x) = x + 2. The inner radius r(x) is the distance from the x-axis to the lower curve y = x, so r(x) = x.
Write the volume integral as: \(V = \pi \int_0^4 \left[ (R(x))^2 - (r(x))^2 \right] \, dx = \pi \int_0^4 \left[ (x+2)^2 - x^2 \right] \, dx\).
Expand the integrand, simplify, and then integrate with respect to x over the interval [0,4] to find the volume of the solid.

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Key Concepts

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

Region Bounded by Curves

The region R is defined by the lines y = x, y = x + 2, x = 0, and x = 4. Understanding how these curves intersect and enclose an area is essential for setting up the integral. The region lies between two parallel lines y = x and y = x + 2, bounded vertically by x = 0 and x = 4.
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Finding Area When Bounds Are Not Given

Volume of Solids of Revolution

When a region is revolved around an axis, it generates a 3D solid. The volume of this solid can be found using methods like the disk/washer or shell method. Here, revolving around the x-axis suggests using washers formed by the vertical distance between the curves.
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Finding Volume Using Disks

Washer Method for Volume Calculation

The washer method involves slicing the solid perpendicular to the axis of revolution, creating washers with an outer and inner radius. The volume is the integral of the area of these washers along the axis. For this problem, the outer radius is y = x + 2 and the inner radius is y = x, integrated from x = 0 to x = 4.
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Finding Volume Using Disks
Related Practice
Textbook Question

45–48. Shell and washer methods about other lines Use both the shell method and the washer method to find the volume of the solid that is generated when the region in the first quadrant bounded by y = x²,y=1, and x=0 is revolved about the following lines. 


x = -1

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Textbook Question

Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis.

y=sin xon [0,π] and y=0 ; about the x-axis (Hint: Recall that sin^2 x=1 − cos2x / 2.

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Textbook Question

Find the area of the region described in the following exercises.


The region bounded by y=4x+4, y=6x+6, and x=4

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Textbook Question

9-34. Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis. 


{Use of Tech} y = 1 / (x² + 1)²,y=0,x=1, and x=2; about the y-axis

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Textbook Question

Find the arc length of the line y = 4−3x on [−3, 2] using calculus and verify your answer using geometry.

Textbook Question

Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given line.


y=x and y=1+x/2; about y=3