1-2/5 1. 'Round and 'round we go (#integration) Our goal in this part is to develop a framework for computing the volume of a specinc class of solids, specifically those that are symmetric and can be constructed as a solid of revolution. We begin by considering a function f(x) in the interval [a, b]. As our first example, let's take the function f(x) = 4-2², in the interval [0, 2], as shown below in Figure 1. The solid of revolution obtained by revolving f(x) around the x-axis is shown in Figure 2. (a) What shape is obtained if we take a vertical slice from the solid of revolution, say at x = 1? What is the area of such slice? How do the shape and its area change if we look at a vertical slice through x = 0? (b) Find a general formula for the area of a vertical slice at any value z. Explain all the terms in your formula.
1-2/5 1. 'Round and 'round we go (#integration) Our goal in this part is to develop a framework for computing the volume of a specinc class of solids, specifically those that are symmetric and can be constructed as a solid of revolution. We begin by considering a function f(x) in the interval [a, b]. As our first example, let's take the function f(x) = 4-2², in the interval [0, 2], as shown below in Figure 1. The solid of revolution obtained by revolving f(x) around the x-axis is shown in Figure 2. (a) What shape is obtained if we take a vertical slice from the solid of revolution, say at x = 1? What is the area of such slice? How do the shape and its area change if we look at a vertical slice through x = 0? (b) Find a general formula for the area of a vertical slice at any value z. Explain all the terms in your formula.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Figure 2: Graph of the solid of revolution obtained by rotating the function f(x) = 4 - x²
in the interval [0, 2] around the x-axis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb6f0fc2b-ed06-42f6-aea2-7d26d8b1334d%2F25c8fabe-4b5f-4443-bb12-52646e2ab634%2Fk42qg2_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Figure 2: Graph of the solid of revolution obtained by rotating the function f(x) = 4 - x²
in the interval [0, 2] around the x-axis.
![Deep Dives
1. 'Round and 'round we go (#integration)
1-2/5
Our goal in this part is to develop a framework for computing the volume of a specinc
class of solids, specifically those that are symmetric and can be constructed as a solid
of revolution. We begin by considering a function f(x) in the interval [a, b]. As our
first example, let's take the function f(x) = 4x², in the interval [0, 2], as shown
below in Figure 1. The solid of revolution obtained by revolving f(x) around the
x-axis is shown in Figure 2.
(a) What shape is obtained if we take a vertical slice from the solid of revolution,
say at x = 1? What is the area of such slice? How do the shape and its area
change if we look at a vertical slice through x = 0?
(b) Find a general formula for the area of a vertical slice at any value x. Explain all
the terms in your formula.
1
(c) We can think of the volume of the solid of revolution as a "stack" of all these
slices. We should thus be able to compute its total volume by summing up the
volume of each of these slices. In order to do that, we must first assign some
thickness to each slice¹, which we will denote Ar. Write an expression for the
volume of:
(i) an individual slice at any value r
(ii) the sum of all slices if we have n total slices.
(d) What limit expression would give us the exact volume of the solid?
(e) Find the integral expression that is equivalent to the limit expression you found
above. Justify your reasoning and explain the values of the bounds on the inte-
gral.
(f) Use this expression to compute the volume of the solid of revolution shown in
Figure 2.
(g) Would the formula change if we had revolved around the y-axis instead? Do you
expect the volume of the solid of revolution obtained by revolving f(x) around
the y-axis to be larger, smaller, or the same as that from part (f)?
(h) [The Method of Disks] Generalize: given any function g(x), find the formula
of the volume of the solid of revolution obtained by revolving g(x) around the (i)
x-axis; (ii) y-axis.
(You may assume g(x) has an inverse g-¹(y)).
Figure 1: Graph of the function f(x)=4-2² in the interval [0, 2].
⠀⠀](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb6f0fc2b-ed06-42f6-aea2-7d26d8b1334d%2F25c8fabe-4b5f-4443-bb12-52646e2ab634%2Fc2mg63b_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Deep Dives
1. 'Round and 'round we go (#integration)
1-2/5
Our goal in this part is to develop a framework for computing the volume of a specinc
class of solids, specifically those that are symmetric and can be constructed as a solid
of revolution. We begin by considering a function f(x) in the interval [a, b]. As our
first example, let's take the function f(x) = 4x², in the interval [0, 2], as shown
below in Figure 1. The solid of revolution obtained by revolving f(x) around the
x-axis is shown in Figure 2.
(a) What shape is obtained if we take a vertical slice from the solid of revolution,
say at x = 1? What is the area of such slice? How do the shape and its area
change if we look at a vertical slice through x = 0?
(b) Find a general formula for the area of a vertical slice at any value x. Explain all
the terms in your formula.
1
(c) We can think of the volume of the solid of revolution as a "stack" of all these
slices. We should thus be able to compute its total volume by summing up the
volume of each of these slices. In order to do that, we must first assign some
thickness to each slice¹, which we will denote Ar. Write an expression for the
volume of:
(i) an individual slice at any value r
(ii) the sum of all slices if we have n total slices.
(d) What limit expression would give us the exact volume of the solid?
(e) Find the integral expression that is equivalent to the limit expression you found
above. Justify your reasoning and explain the values of the bounds on the inte-
gral.
(f) Use this expression to compute the volume of the solid of revolution shown in
Figure 2.
(g) Would the formula change if we had revolved around the y-axis instead? Do you
expect the volume of the solid of revolution obtained by revolving f(x) around
the y-axis to be larger, smaller, or the same as that from part (f)?
(h) [The Method of Disks] Generalize: given any function g(x), find the formula
of the volume of the solid of revolution obtained by revolving g(x) around the (i)
x-axis; (ii) y-axis.
(You may assume g(x) has an inverse g-¹(y)).
Figure 1: Graph of the function f(x)=4-2² in the interval [0, 2].
⠀⠀
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