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) = 4x², 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. (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 slice³, 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. (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)? Figure 3: Graph of the solid of revolution obtained by rotating the function f(x)=4-² in the interval [0, 2] around the y-axis. Figure 1: Graph of the function f(x)=4-2² in the interval [0, 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) = 4x², 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. (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 slice³, 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. (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)? Figure 3: Graph of the solid of revolution obtained by rotating the function f(x)=4-² in the interval [0, 2] around the y-axis. Figure 1: Graph of the function f(x)=4-2² in the interval [0, 2]
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter8: Further Techniques And Applications Of Integration
Section8.3: Volume And Average Value
Problem 22E
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Question
![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- r², 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.
(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 slice³, which we will denote Az. Write an
expression for the volume of:
(i) an individual slice of radius æ;
(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)?
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) = 4-2² in the interval [0, 2]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F03b5f08a-ed30-4408-a657-de0de7174de5%2F17b8f2a2-0fa5-48fc-bf46-270fa4ce55fd%2F9fxkx69_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- r², 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.
(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 slice³, which we will denote Az. Write an
expression for the volume of:
(i) an individual slice of radius æ;
(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)?
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) = 4-2² in the interval [0, 2]
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