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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 6.3.40
Chapter 6, Problem 6.3.40

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^2,y=2−x, and y=0; about the y-axis

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First, identify the region R bounded by the curves: \(y = x^2\), \(y = 2 - x\), and \(y = 0\). Sketching these curves helps visualize the area and the limits of integration.
Find the points of intersection between the curves to determine the bounds for \(x\). Solve \(x^2 = 2 - x\) to find the \(x\)-values where the curves \(y = x^2\) and \(y = 2 - x\) intersect.
Since the solid is generated by revolving the region around the y-axis, use the method of cylindrical shells. The volume element for shells is given by \(dV = 2\pi \cdot (\text{radius}) \cdot (\text{height}) \cdot dx\).
Express the radius and height of a typical shell in terms of \(x\). The radius is the distance from the y-axis, which is \(x\), and the height is the vertical distance between the curves, which is \((2 - x) - x^2\).
Set up the integral for the volume as \(V = \int_{a}^{b} 2\pi x \left[(2 - x) - x^2\right] dx\), where \(a\) and \(b\) are the \(x\)-values found from the intersection points. This integral will give 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.

Finding the Region Bounded by Curves

To find the volume of a solid generated by revolving a region, first identify the area bounded by the given curves. This involves determining the points of intersection and understanding the shape enclosed by y = x², y = 2 - x, and y = 0. Accurate sketching or solving equations helps visualize the region R.
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Finding Area When Bounds Are Not Given

Method of Cylindrical Shells

When revolving a region around the y-axis, the cylindrical shells method is often used. It involves integrating the volume of thin cylindrical shells with radius equal to the x-value, height given by the difference of functions, and thickness dx. This method simplifies volume calculation when the axis of rotation is vertical.
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Euler's Method

Setting up and Evaluating Definite Integrals

Calculating the volume requires setting up a definite integral with proper limits corresponding to the region's x-values. The integral sums the volumes of infinitesimal shells or disks. Correctly expressing the radius, height, and limits ensures accurate evaluation of the volume.
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Definition of the Definite Integral
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 line.


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

Textbook Question

Look again at the region R in Figure 6.38 (p. 439). Explain why it would be difficult to use the washer method to find the volume of the solid of revolution that results when R is revolved about the y-axis.

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Textbook Question

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

y = 3 ln x− x²/24 on [1, 6]

Textbook Question

Suppose the region bounded by the curve y=f(x) from x=0 to x=4 (see figure) is revolved about the x-axis to form a solid of revolution. Use left, right, and midpoint Riemann sums, with n=4 subintervals of equal length, to estimate the volume of the solid of revolution.

Textbook Question

Use the general slicing method to find the volume of the following solids.

The solid whose base is the region bounded by the curves y=x^2 and y=2−x^2, and whose cross sections through the solid perpendicular to the x-axis are squares

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