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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 6.3.48
Chapter 6, Problem 6.3.48

For the following regions R, determine which is greater—the volume of the solid generated when R is revolved about the x-axis or about the y-axis.


R is bounded by y=x^2 and y=√8x.

Verified step by step guidance
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First, identify the region R bounded by the curves \(y = x^2\) and \(y = \sqrt{8x}\). To do this, find the points of intersection by setting \(x^2 = \sqrt{8x}\).
Solve the equation \(x^2 = \sqrt{8x}\) by squaring both sides to eliminate the square root, resulting in \(x^4 = 8x\). Then, rearrange to \(x^4 - 8x = 0\) and factor to find the intersection points.
Determine the limits of integration (the x-values where the curves intersect) to define the interval over which the region R lies.
Calculate the volume of the solid generated by revolving R about the x-axis using the disk/washer method. The volume is given by \(V_x = \pi \int_{a}^{b} \left( (\text{outer radius})^2 - (\text{inner radius})^2 \right) \, dx\), where the radii are the distances from the x-axis to the curves.
Calculate the volume of the solid generated by revolving R about the y-axis using the shell method. The volume is given by \(V_y = 2\pi \int_{a}^{b} (\text{radius})(\text{height}) \, dx\), where the radius is the distance from the y-axis and the height is the vertical distance between the curves.

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Key Concepts

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

Finding the Region of Integration

To analyze the volume generated by revolving a region, first identify the area bounded by the given curves. Here, the region R is enclosed by y = x² and y = √(8x). Finding their points of intersection determines the limits of integration, which is essential for setting up volume integrals.
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Area of Polar Regions

Volume of Solids of Revolution Using the Disk/Washer Method

The disk/washer method calculates volume by integrating cross-sectional areas perpendicular to the axis of revolution. When revolving around the x-axis, slices are horizontal, and when revolving around the y-axis, slices are vertical. Setting up the correct integral expressions depends on the axis chosen.
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Finding Volume Using Disks

Comparing Volumes Generated by Different Axes of Revolution

Volumes generated by revolving the same region about different axes can differ significantly. Understanding how the shape and orientation of the region relative to the axis affect the radius and height of disks or washers helps compare which volume is larger without necessarily computing exact values.
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Finding Volume Using Disks Example 4
Related Practice
Textbook Question

9–20. Arc length calculations Find the arc length of the following curves on the given interval.

y = (x²+2)^3/2 / 3 on [0, 1]

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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.

y = 1−x²,x = 0, and y = 0, in the first quadrant; about the y-axis

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Textbook Question

A 1.5-mm layer of paint is applied to one side of the following surfaces. Find the approximate volume of paint needed. Assume x and y are measured in meters. 


The spherical zone generated when the curve y=√8x−x^2 on the interval 1≤x≤7 is revolved about the x-axis 

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Textbook Question

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


The region bounded by y=2−|x|and y=x^2

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=0,y=lnx,y=2, and x=0; about the y-axis

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Textbook Question

Let f(x) = {x   if 0≤x≤2

      2x−2  if 2<x≤5

      −2x+18 if 5<x≤6. 


Find the volume of the solid formed when the region bounded by the graph of f, the x-axis, and the line x=6 is revolved about the x-axis.