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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 6.3.58
Chapter 6, Problem 6.3.58

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

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First, identify the region R bounded by the curves \( y = x \) and \( y = 1 + \frac{x}{2} \). To do this, find the points of intersection by setting the two equations equal: \( x = 1 + \frac{x}{2} \).
Solve the equation \( x = 1 + \frac{x}{2} \) for \( x \) to find the limits of integration. This will give the \( x \)-values where the two curves intersect, which define the interval over which the region R lies.
Since the solid is generated by revolving the region about the line \( y = 3 \), use the washer method. The radius of each washer is the distance from the line \( y = 3 \) to the curve. Express the outer radius \( R(x) \) and inner radius \( r(x) \) as functions of \( x \): \[ R(x) = 3 - y_{\text{lower}}(x), \quad r(x) = 3 - y_{\text{upper}}(x) \] where \( y_{\text{lower}}(x) \) and \( y_{\text{upper}}(x) \) are the lower and upper curves respectively on the interval.
Set up the volume integral using the washer method formula: \[ V = \pi \int_{a}^{b} \left[ R(x)^2 - r(x)^2 \right] \, dx \] where \( a \) and \( b \) are the intersection points found in step 2.
Evaluate the integral to find the volume of the solid. This involves squaring the radii expressions, subtracting, and integrating with respect to \( x \) over the interval \( [a, b] \).

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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 solving for the points of intersection and understanding which curve lies above or below within the interval. Here, the region is bounded by y = x and y = 1 + x/2.
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Finding Area When Bounds Are Not Given

Volume of Solids of Revolution Using the Washer Method

When a region is revolved around a horizontal line not on the x-axis, the volume can be found using the washer method. This involves integrating the difference of the squares of the outer and inner radii (distances from the axis of rotation to the curves) with respect to x or y, depending on the setup.
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Finding Volume Using Disks

Adjusting Radii for Revolution About a Line y = k

When revolving around a horizontal line y = k, the radius of each washer is the vertical distance between the curve and the line y = k. This requires calculating |k - y| for each curve to determine the inner and outer radii, ensuring correct setup of the integral for volume.
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Equations of Tangent Lines
Related Practice
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=x^2,y=2−x, and y=0; about the y-axis

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

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

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

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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