Preview Activity 11.1.1. In this activity we introduce the concept of a double Riemann sum. ………….. a. Review the concept of the Riemann sum from single-variable calculus. Then, explain how we define the definite integral f(x) dx of a continuous function of a single variable z on an interval [a, b]. Include a sketch of a continuous function on an interval [a,b] with appropriate labeling in order to illustrate your definition. b. In our upcoming study of integral calculus for multivariable functions, we will first extend the idea of the single-variable definite integral to functions of two variables over rectangular domains. To do so, we will need to understand how to partition a rectangle into subrectangles. Let R be rectangular domain R = {(z,y): 0≤x≤6,2 ≤ y ≤ 4} (we can also represent this domain with the notation [0, 6] x [2, 4]), as pictured in Figure 11.1.1. 2 Figure 11.1.1. Rectangular domain R with subrectangles. To form a partition of the full rectangular region, R, we will partition both intervals [0,6] and [2, 4]; in particular, we choose to partition the interval [0,6] into three uniformly sized subintervals and the interval [2, 4] into two evenly sized subintervals as shown in Figure 11.1.1. In the following questions, we discuss how to identify the endpoints of each subinterval and the resulting subrectangles. i. Let 0 = 20 <1 < 2 <3 = 6 be the endpoints of the subintervals of [0, 6] after partitioning. What is the length Az of each subinterval [₁-1,2₁] for i from 1 to 3? ii. Explicitly identify 10, 1₁, 12, and 13. On Figure 11.1.1 or your own version of the diagram, label these endpoints. iii. Let 2 = Yo < y₁ < y2 = 4 be the endpoints of the subintervals of [2, 4] after partitioning. What is the length Ay of each subinterval [y₁-1, y;] for j from 1 to 2? Identify 30, 3₁, and 32 and label these endpoints on Figure 11.1.1. iv. Let R₁; denote the subrectangle [₁-1,₁] × [y₁-1, y;]. Appropriately label each subrectangle in your drawing of Figure 11.1.1. How does the total number of subrectangles depend on the partitions of the intervals [0, 6] and [2, 4]? v. What is area AA of each subrectangle?
Preview Activity 11.1.1. In this activity we introduce the concept of a double Riemann sum. ………….. a. Review the concept of the Riemann sum from single-variable calculus. Then, explain how we define the definite integral f(x) dx of a continuous function of a single variable z on an interval [a, b]. Include a sketch of a continuous function on an interval [a,b] with appropriate labeling in order to illustrate your definition. b. In our upcoming study of integral calculus for multivariable functions, we will first extend the idea of the single-variable definite integral to functions of two variables over rectangular domains. To do so, we will need to understand how to partition a rectangle into subrectangles. Let R be rectangular domain R = {(z,y): 0≤x≤6,2 ≤ y ≤ 4} (we can also represent this domain with the notation [0, 6] x [2, 4]), as pictured in Figure 11.1.1. 2 Figure 11.1.1. Rectangular domain R with subrectangles. To form a partition of the full rectangular region, R, we will partition both intervals [0,6] and [2, 4]; in particular, we choose to partition the interval [0,6] into three uniformly sized subintervals and the interval [2, 4] into two evenly sized subintervals as shown in Figure 11.1.1. In the following questions, we discuss how to identify the endpoints of each subinterval and the resulting subrectangles. i. Let 0 = 20 <1 < 2 <3 = 6 be the endpoints of the subintervals of [0, 6] after partitioning. What is the length Az of each subinterval [₁-1,2₁] for i from 1 to 3? ii. Explicitly identify 10, 1₁, 12, and 13. On Figure 11.1.1 or your own version of the diagram, label these endpoints. iii. Let 2 = Yo < y₁ < y2 = 4 be the endpoints of the subintervals of [2, 4] after partitioning. What is the length Ay of each subinterval [y₁-1, y;] for j from 1 to 2? Identify 30, 3₁, and 32 and label these endpoints on Figure 11.1.1. iv. Let R₁; denote the subrectangle [₁-1,₁] × [y₁-1, y;]. Appropriately label each subrectangle in your drawing of Figure 11.1.1. How does the total number of subrectangles depend on the partitions of the intervals [0, 6] and [2, 4]? v. What is area AA of each subrectangle?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![Preview Activity 11.1.1. In this activity we introduce the concept of a
double Riemann sum.
…………..
a. Review the concept of the Riemann sum from single-variable calculus.
Then, explain how we define the definite integral ff(x) dz of a
continuous function of a single variable z on an interval [a, b]. Include a
sketch of a continuous function on an interval [a,b] with appropriate
labeling in order to illustrate your definition.
b. In our upcoming study of integral calculus for multivariable functions, we
will first extend the idea of the single-variable definite integral to
functions of two variables over rectangular domains. To do so, we will
need to understand how to partition a rectangle into subrectangles. Let
R be rectangular domain R = {(z,y): 0≤x≤6,2 ≤ y ≤ 4} (we can also
represent this domain with the notation [0, 6] x [2, 4]), as pictured in
Figure 11.1.1.
2
Figure 11.1.1. Rectangular domain R with subrectangles.
To form a partition of the full rectangular region, R, we will partition both
intervals [0,6] and [2, 4]; in particular, we choose to partition the interval
[0, 6] into three uniformly sized subintervals and the interval [2, 4] into
two evenly sized subintervals as shown in Figure 11.1.1. In the following
questions, we discuss how to identify the endpoints of each subinterval
and the resulting subrectangles.
i. Let 0 = 20 <1 < 2 <3 = 6 be the endpoints of the subintervals
of [0, 6] after partitioning. What is the length Az of each subinterval
[₁-1,2₁] for i from 1 to 3?
ii. Explicitly identify 10, 1₁, 12, and 13. On Figure 11.1.1 or your own
version of the diagram, label these endpoints.
iii. Let 2 = Yo < y₁ < y2 = 4 be the endpoints of the subintervals of
[2, 4] after partitioning. What is the length Ay of each subinterval
[y₁-1, y;] for j from 1 to 2? Identify 30, 3₁, and 32 and label these
endpoints on Figure 11.1.1.
iv. Let R₁; denote the subrectangle [₁-1,₁] × [y₁-1, y;]. Appropriately
label each subrectangle in your drawing of Figure 11.1.1. How does
the total number of subrectangles depend on the partitions of the
intervals [0, 6] and [2, 4]?
v. What is area AA of each subrectangle?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F409330f8-4765-4598-8146-83752bc8a849%2Fd573fc18-f695-4023-a730-e4e369b72f9d%2Fax81zo_processed.png&w=3840&q=75)
Transcribed Image Text:Preview Activity 11.1.1. In this activity we introduce the concept of a
double Riemann sum.
…………..
a. Review the concept of the Riemann sum from single-variable calculus.
Then, explain how we define the definite integral ff(x) dz of a
continuous function of a single variable z on an interval [a, b]. Include a
sketch of a continuous function on an interval [a,b] with appropriate
labeling in order to illustrate your definition.
b. In our upcoming study of integral calculus for multivariable functions, we
will first extend the idea of the single-variable definite integral to
functions of two variables over rectangular domains. To do so, we will
need to understand how to partition a rectangle into subrectangles. Let
R be rectangular domain R = {(z,y): 0≤x≤6,2 ≤ y ≤ 4} (we can also
represent this domain with the notation [0, 6] x [2, 4]), as pictured in
Figure 11.1.1.
2
Figure 11.1.1. Rectangular domain R with subrectangles.
To form a partition of the full rectangular region, R, we will partition both
intervals [0,6] and [2, 4]; in particular, we choose to partition the interval
[0, 6] into three uniformly sized subintervals and the interval [2, 4] into
two evenly sized subintervals as shown in Figure 11.1.1. In the following
questions, we discuss how to identify the endpoints of each subinterval
and the resulting subrectangles.
i. Let 0 = 20 <1 < 2 <3 = 6 be the endpoints of the subintervals
of [0, 6] after partitioning. What is the length Az of each subinterval
[₁-1,2₁] for i from 1 to 3?
ii. Explicitly identify 10, 1₁, 12, and 13. On Figure 11.1.1 or your own
version of the diagram, label these endpoints.
iii. Let 2 = Yo < y₁ < y2 = 4 be the endpoints of the subintervals of
[2, 4] after partitioning. What is the length Ay of each subinterval
[y₁-1, y;] for j from 1 to 2? Identify 30, 3₁, and 32 and label these
endpoints on Figure 11.1.1.
iv. Let R₁; denote the subrectangle [₁-1,₁] × [y₁-1, y;]. Appropriately
label each subrectangle in your drawing of Figure 11.1.1. How does
the total number of subrectangles depend on the partitions of the
intervals [0, 6] and [2, 4]?
v. What is area AA of each subrectangle?
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