TBIL-Calc Volumes of Revolution (AI3)

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Section 6.3 Volumes of Revolution (AI3)

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Learning Outcomes

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Compute volumes of solids of revolution.

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Subsection 6.3.1 Activities

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Activity 6.3.1.

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Consider the following visualization to decide which of these statements is most appropriate for describing the relationship of lengths and areas.


    
        
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f(x) = (x-3)*(x-5)*(x-7)+40
2
p = line([(2,0),(2,f(2))], color='black')
3
p += line([(8,0),(8,f(8))], color='black')
4
p += polygon([(2,0),(2,f(2))] + [(x, f(x)) for x in [2,2.1,..,8]] + [(8,0),(2,0)],  rgbcolor=(0.8,0.8,0.8),aspect_ratio='automatic')
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p += text("$\\mathrm{Area}=\\int_{a}^b f(x) dx$", (4, 55), fontsize=16, color='black')
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p += plot(f, (1, 8.5), thickness=3)
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p += line([(6,0),(6,f(6))], color="#8888ff")
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p += text("$\\mathrm{Length}=f(x)$", (5.9, 20), fontsize=16, color='black', horizontal_alignment='right')
9
p

    
    
    
    
        
            
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        Messages

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Length is the integral of areas.
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Area is the integral of lengths.
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Length is the derivative of areas.
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None of these.

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Definition 6.3.2.

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We define the volume of a solid with cross sectional area given by

A (x)

laying between

a \leq x \leq b

to be the definite integral

Volume = \int_{a}^{b} A (x) d x .


    
        
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x,y,z=var("x y z")
2
p = revolution_plot3d(x/2,(x,0,4), show_curve=False,parallel_axis="x",opacity=0.5,frame=False)
3
p += revolution_plot3d((4,z),(z,0,2), show_curve=False,parallel_axis="x",opacity=0.5)
4
p += revolution_plot3d((2,z),(z,0,1), show_curve=False,parallel_axis="x",opacity=1)
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p += line([(0,0,-3), (0,0,3)],color="black",label="x")
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p += line([(-1,0,0), (5,0,0)],color="black")
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p += text3d("Volume=∫A(x)dx",(4,0,2.5))
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p += text3d("Area=A(x)",(2,0,1.25))
9
p

    
    
    
    
        
            
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        Messages

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Activity 6.3.3.

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We will be focused on the volumes of solids obtained by revolving a region around an axis. Let's use the running example of the region bounded by the curves

x = 0, y = 4, y = x^{2} .


    
        
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x,y,z = var('x y z')
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p = plot(x^2,(x,0,2),aspect_ratio=1,color='red')
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p += line([(0,0),(0,4),(2,4)],color="red")
4
p

    
    
    
    
        
            
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        Messages

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(a)

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Consider the below illustrated revolution of this region, and the cross-section drawn from a horizontal line segment. Choose the most appropriate description of this illustration.


    
        
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1
x,y,z = var('x y z')
2
p = revolution_plot3d(x^2,(x,0,2), show_curve=True,parallel_axis="z",opacity=0.5,frame=False)
3
p += revolution_plot3d(4,(x,0,2), show_curve=True,parallel_axis="z",opacity=0.5)
4
p += revolution_plot3d(2,(x,0,sqrt(2)), show_curve=True,parallel_axis="z",opacity=1)
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p += line([(0,0,-1), (0,0,5)],color="black",label="x")
6
p += line([(-4,0,0), (4,0,0)],color="black")
7
p

    
    
    
    
        
            
                Language:
                
            
        
    
    




    
    
        
        Messages

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Region is rotated around the $x$ -axis; the cross-sectional area is determined by the line segment's $x$ -value.
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Region is rotated around the $x$ -axis; the cross-sectional area is determined by the line segment's $y$ -value.
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Region is rotated around the $y$ -axis; the cross-sectional area is determined by the line segment's $x$ -value.
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Region is rotated around the $y$ -axis; the cross-sectional area is determined by the line segment's $y$ -value.

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(b)

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Which of these formulas is most appropriate to find this illustration's cross-sectional area?

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$π r^{2}$
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$2 π r h$
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$π R^{2} - π r^{2}$
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$\frac{1}{2} b h$

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(c)

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Consider the below illustrated revolution of this region, and the cross-section drawn from a vertical line segment. Choose the most appropriate description of this illustration.


    
        
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1
x,y,z = var('x y z')
2
p = revolution_plot3d(x^2,(x,0,2), show_curve=True,parallel_axis="z",opacity=0.5,frame=False)
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p += revolution_plot3d(4,(x,0,2), show_curve=True,parallel_axis="z",opacity=0.5)
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p += revolution_plot3d((1,x),(x,1,4), show_curve=True,parallel_axis="z",opacity=1)
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p += line([(0,0,-1), (0,0,5)],color="black")
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p += line([(-4,0,0), (4,0,0)],color="black")
7
p

    
    
    
    
        
            
                Language:
                
            
        
    
    




    
    
        
        Messages

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Region is rotated around the $x$ -axis; the cross-sectional area is determined by the line segment's $x$ -value.
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Region is rotated around the $x$ -axis; the cross-sectional area is determined by the line segment's $y$ -value.
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Region is rotated around the $y$ -axis; the cross-sectional area is determined by the line segment's $x$ -value.
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Region is rotated around the $y$ -axis; the cross-sectional area is determined by the line segment's $y$ -value.

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(d)

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Which of these formulas is most appropriate to find this illustration's cross-sectional area?

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$π r^{2}$
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$2 π r h$
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$π R^{2} - π r^{2}$
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$\frac{1}{2} b h$

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(e)

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Consider the below illustrated revolution of this region, and the cross-section drawn from a horizontal line segment. Choose the most appropriate description of this illustration.


    
        
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1
x,y,z = var('x y z')
2
p = revolution_plot3d(x^2,(x,0,2), show_curve=True,parallel_axis="x",opacity=0.5,frame=False)
3
p += revolution_plot3d(4,(x,0,2), show_curve=True,parallel_axis="x",opacity=0.5)
4
p += revolution_plot3d((0,z),(z,0,4), show_curve=True,parallel_axis="x",opacity=0.5)
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p += revolution_plot3d(2,(x,0,sqrt(2)), show_curve=True,parallel_axis="x",opacity=1)
6
p += line([(0,0,-5), (0,0,5)],color="black",label="x")
7
p += line([(-1,0,0), (4,0,0)],color="black")
8
p

    
    
    
    
        
            
                Language:
                
            
        
    
    




    
    
        
        Messages

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Region is rotated around the $x$ -axis; the cross-sectional area is determined by the line segment's $x$ -value.
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Region is rotated around the $x$ -axis; the cross-sectional area is determined by the line segment's $y$ -value.
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Region is rotated around the $y$ -axis; the cross-sectional area is determined by the line segment's $x$ -value.
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Region is rotated around the $y$ -axis; the cross-sectional area is determined by the line segment's $y$ -value.

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(f)

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Which of these formulas is most appropriate to find this illustration's cross-sectional area?

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$π r^{2}$
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$2 π r h$
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$π R^{2} - π r^{2}$
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$\frac{1}{2} b h$

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(g)

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Consider the below illustrated revolution of this region, and the cross-section drawn from a vertical line segment. Choose the most appropriate description of this illustration.


    
        
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1
x,y,z = var('x y z')
2
p = revolution_plot3d(x^2,(x,0,2), show_curve=True,parallel_axis="x",opacity=0.5,frame=False)
3
p += revolution_plot3d(4,(x,0,2), show_curve=True,parallel_axis="x",opacity=0.5)
4
p += revolution_plot3d((0,z),(z,0,4), show_curve=True,parallel_axis="x",opacity=0.5)
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p += revolution_plot3d((1,x),(x,1,4), show_curve=True,parallel_axis="x",opacity=1)
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p += line([(0,0,-5), (0,0,5)],color="black",label="x")
7
p += line([(-1,0,0), (4,0,0)],color="black")
8
p

    
    
    
    
        
            
                Language:
                
            
        
    
    




    
    
        
        Messages

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Region is rotated around the $x$ -axis; the cross-sectional area is determined by the line segment's $x$ -value.
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Region is rotated around the $x$ -axis; the cross-sectional area is determined by the line segment's $y$ -value.
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Region is rotated around the $y$ -axis; the cross-sectional area is determined by the line segment's $x$ -value.
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Region is rotated around the $y$ -axis; the cross-sectional area is determined by the line segment's $y$ -value.

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(h)

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Which of these formulas is most appropriate to find this illustration's cross-sectional area?

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$π r^{2}$
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$2 π r h$
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$π R^{2} - π r^{2}$
🔗
$\frac{1}{2} b h$

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Remark 6.3.4.

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Generally when solving problems without the aid of technology, it's useful to draw your region in two dimensions, choose whether to use a horizontal or vertical line segment, and draw its rotation to determine the cross-sectional shape.

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When the shape is a disk, this is called the disk method and we use one of these formulas depending on whether the cross-sectional area depends on

x

y .

V = \int_{a}^{b} π r (x)^{2} d x, V = \int_{a}^{b} π r (y)^{2} d y .

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When the shape is a washer, this is called the washer method and we use one of these formulas depending on whether the cross-sectional area depends on

x

y .

V = \int_{a}^{b} (π R (x)^{2} - π r (x)^{2}) d x, V = \int_{a}^{b} (π R (y)^{2} - π r (y)^{2}) d y .

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When the shape is a cylindrical shell, this is called the shell method and we use one of these formulas depending on whether the cross-sectional area depends on