(a) Say we take two circular cookie cutter, one of radius r₁ = 1 and the other with radius 72 = 1 + Ar, and use them to cut out a slice of our solid of revolution. What is the 3-dimensional shape of the slice? What is the surface area of this shape? (b) Find a general formula for the surface area of the slice obtained by cutting the solid with cookie cutters of radii r₁ = x and r2 = x + Ax. Explain all the terms in your formula.² lere, there are technically two areas that you can look at, an inner and an outer area. You may pick , but just note: what happens to those areas in the limit as Ar tends to zero? 3 (c) We can think of the volume of the solid of revolution as a collection of slices of cookie cutters of many radii. 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 slice, which we will denote Ar. Write an expression for the volume of: (i) an individual slice of radius ; (ii) the sum of all slices if we have n total slices.
(a) Say we take two circular cookie cutter, one of radius r₁ = 1 and the other with radius 72 = 1 + Ar, and use them to cut out a slice of our solid of revolution. What is the 3-dimensional shape of the slice? What is the surface area of this shape? (b) Find a general formula for the surface area of the slice obtained by cutting the solid with cookie cutters of radii r₁ = x and r2 = x + Ax. Explain all the terms in your formula.² lere, there are technically two areas that you can look at, an inner and an outer area. You may pick , but just note: what happens to those areas in the limit as Ar tends to zero? 3 (c) We can think of the volume of the solid of revolution as a collection of slices of cookie cutters of many radii. 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 slice, which we will denote Ar. Write an expression for the volume of: (i) an individual slice of radius ; (ii) the sum of all slices if we have n total slices.
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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![2. Cookie-cutting (#integration)
In the previous part, you developed the Method of Disks for computing the volume
of solids of revolution. We will now find an alternative method, the Method of
Shells. We will consider the same function f(x)=4-2², in the interval [0, 2], as
shown in Figure 1, but now, we will first consider the solid of revolution obtained by
revolving f(x) around the y-axis is shown in Figure 3.
(a) Say we take two circular cookie cutter, one of radius r₁= 1 and the other with
radius r2 = 1 + Ar, and use them to cut out a slice of our solid of revolution.
What is the 3-dimensional shape of the slice? What is the surface area of this
shape?
(b) Find a general formula for the surface area of the slice obtained by cutting the
solid with cookie cutters of radii r₁ = x and r₂ = x + Ar. Explain all the terms
in your formula.²
2Here, there are technically two areas that you can look at, an inner and an outer area. You may pick
either, but just note: what happens to those areas in the limit as Ar tends to zero?
3
(c) We can think of the volume of the solid of revolution as a collection of slices of
cookie cutters of many radii. 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 slice, which we will denote Az. Write an
expression for the volume of:
(i) an individual slice of radius x;
(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. Use this expression to compute the volume of
the solid of revolution shown in Figure 3. How does your answer compare with
problem 1(g)?
(f) Would the formula change if we had revolved around the z-axis instead?
(g) [The Method of Shells] Generalize: given any function g(x), use the Method
of Shells to find the formula of the volume of the solid of revolution obtained by
revolving g(x) around the (i) y-axis; (ii) z-axis.
(You may assume g(x) has an inverse g-¹(y)).
(h) From your experience in these first two problems, which method yielded a simpler
calculation? What do you think are the best uses of each method?
(i) Discuss specific cases of solids of revolution in real life whose volumes could be
computed by the methods described here.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb6f0fc2b-ed06-42f6-aea2-7d26d8b1334d%2F87b32ee0-2cce-4f93-927b-2cb33f2bbb01%2F5zakdbp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2. Cookie-cutting (#integration)
In the previous part, you developed the Method of Disks for computing the volume
of solids of revolution. We will now find an alternative method, the Method of
Shells. We will consider the same function f(x)=4-2², in the interval [0, 2], as
shown in Figure 1, but now, we will first consider the solid of revolution obtained by
revolving f(x) around the y-axis is shown in Figure 3.
(a) Say we take two circular cookie cutter, one of radius r₁= 1 and the other with
radius r2 = 1 + Ar, and use them to cut out a slice of our solid of revolution.
What is the 3-dimensional shape of the slice? What is the surface area of this
shape?
(b) Find a general formula for the surface area of the slice obtained by cutting the
solid with cookie cutters of radii r₁ = x and r₂ = x + Ar. Explain all the terms
in your formula.²
2Here, there are technically two areas that you can look at, an inner and an outer area. You may pick
either, but just note: what happens to those areas in the limit as Ar tends to zero?
3
(c) We can think of the volume of the solid of revolution as a collection of slices of
cookie cutters of many radii. 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 slice, which we will denote Az. Write an
expression for the volume of:
(i) an individual slice of radius x;
(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. Use this expression to compute the volume of
the solid of revolution shown in Figure 3. How does your answer compare with
problem 1(g)?
(f) Would the formula change if we had revolved around the z-axis instead?
(g) [The Method of Shells] Generalize: given any function g(x), use the Method
of Shells to find the formula of the volume of the solid of revolution obtained by
revolving g(x) around the (i) y-axis; (ii) z-axis.
(You may assume g(x) has an inverse g-¹(y)).
(h) From your experience in these first two problems, which method yielded a simpler
calculation? What do you think are the best uses of each method?
(i) Discuss specific cases of solids of revolution in real life whose volumes could be
computed by the methods described here.
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