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 - x², 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 = r + Ar. Explain all the terms in your formula.²
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 - x², 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 = r + Ar. Explain all the terms in your formula.²
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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Solve by hand. Explain and show calculations by steps. Handwritten.
![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) = 4-², 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 =r + Ar. Explain all the terms
in your formula.²
Figure 3: Graph of the solid of revolution obtained by rotating the function f(x) = 4 - x²|
in the interval [0, 2] around the y-axis.
Figure 1: Graph of the function f(x) = 4x² in the interval [0, 2]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F03b5f08a-ed30-4408-a657-de0de7174de5%2F319925f5-f2b7-47cf-b316-0c27b7a24fbb%2Fg2ozsek_processed.png&w=3840&q=75)
Transcribed Image Text: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) = 4-², 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 =r + Ar. Explain all the terms
in your formula.²
Figure 3: Graph of the solid of revolution obtained by rotating the function f(x) = 4 - x²|
in the interval [0, 2] around the y-axis.
Figure 1: Graph of the function f(x) = 4x² in the interval [0, 2]
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