Preview Activity 11.2.1. Let f(x, y) = 25 – x² — y² on the rectangular domain R= [-3, 3] x [-4, 4]. As with partial derivatives, we may treat one of the variables in f as constant and think of the resulting function as a function of a single variable. Now we investigate what happens if we integrate instead of differentiate. a. Choose a fixed value of x in the interior of [-3,3]. Let What is the geometric meaning of the value of A(x) relative to the surface defined by f. (Hint: Think about the trace determined by the fixed value of x, and consider how A(x) is related to the image at left in Figure 11.2.2.) -4 -2 0 Y 4 A(z) = ª* f(x, y) dy. -4 2 X -1 Z 25 20 F15 10 +5 -4 -2 0 Y 2 X Z -3 25 20 15 10 5 Figure 11.2.2. Left: A cross section with fixed x. Right: A cross section with fixed x and Ax. X b. For a fixed value of x, say x, what is the geometric meaning of A(x) Ax? (Hint: Consider how A(x*)Aï is related to the image at right in Figure 11.2.2.) b. For a fixed value of ï, say ¤‡, what is the geometric meaning of A(x) Ax? (Hint: Consider how A(**)Ax is related to the image at right in Figure 11.2.2.) c. Since f is continuous on R, we can define the function A = A(x) at every value of x in [−3, 3]. Now think about subdividing the x-interval [-3,3] into m subintervals, and choosing a value * in each of those subintervals. What will be the meaning of the sum ₁ A(x*) Ax? Im i=1 -3 d. Explain why ³ A(x) dx will determine the exact value of the volume under the surface z = f(x, y) over the rectangle R.
Preview Activity 11.2.1. Let f(x, y) = 25 – x² — y² on the rectangular domain R= [-3, 3] x [-4, 4]. As with partial derivatives, we may treat one of the variables in f as constant and think of the resulting function as a function of a single variable. Now we investigate what happens if we integrate instead of differentiate. a. Choose a fixed value of x in the interior of [-3,3]. Let What is the geometric meaning of the value of A(x) relative to the surface defined by f. (Hint: Think about the trace determined by the fixed value of x, and consider how A(x) is related to the image at left in Figure 11.2.2.) -4 -2 0 Y 4 A(z) = ª* f(x, y) dy. -4 2 X -1 Z 25 20 F15 10 +5 -4 -2 0 Y 2 X Z -3 25 20 15 10 5 Figure 11.2.2. Left: A cross section with fixed x. Right: A cross section with fixed x and Ax. X b. For a fixed value of x, say x, what is the geometric meaning of A(x) Ax? (Hint: Consider how A(x*)Aï is related to the image at right in Figure 11.2.2.) b. For a fixed value of ï, say ¤‡, what is the geometric meaning of A(x) Ax? (Hint: Consider how A(**)Ax is related to the image at right in Figure 11.2.2.) c. Since f is continuous on R, we can define the function A = A(x) at every value of x in [−3, 3]. Now think about subdividing the x-interval [-3,3] into m subintervals, and choosing a value * in each of those subintervals. What will be the meaning of the sum ₁ A(x*) Ax? Im i=1 -3 d. Explain why ³ A(x) dx will determine the exact value of the volume under the surface z = f(x, y) over the rectangle R.
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.
Chapter3: The Derivative
Section3.2: Continuity
Problem 36E
Related questions
Question
![Preview Activity 11.2.1. Let f(x, y) = 25 – x² — y² on the rectangular
domain R= [-3, 3] x [-4, 4].
As with partial derivatives, we may treat one of the variables in f as constant
and think of the resulting function as a function of a single variable. Now we
investigate what happens if we integrate instead of differentiate.
a.
Choose a fixed value of x in the interior of [-3,3]. Let
What is the geometric meaning of the value of A(x) relative to the
surface defined by f. (Hint: Think about the trace determined by the fixed
value of x, and consider how A(x) is related to the image at left in
Figure 11.2.2.)
-4
-2 0
Y
4
A(z) = ª* f(x, y) dy.
-4
2
X
-1
Z 25
20
F15
10
+5
-4 -2 0
Y
2
X
Z
-3
25
20
15
10
5
Figure 11.2.2. Left: A cross section with fixed x. Right: A cross section with
fixed x and Ax.
X
b. For a fixed value of x, say x, what is the geometric meaning of
A(x) Ax? (Hint: Consider how A(x*)Aï is related to the image at right
in Figure 11.2.2.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F409330f8-4765-4598-8146-83752bc8a849%2F8f984369-035f-4684-b6ba-ad49b4c1cc25%2Fm9axege_processed.png&w=3840&q=75)
Transcribed Image Text:Preview Activity 11.2.1. Let f(x, y) = 25 – x² — y² on the rectangular
domain R= [-3, 3] x [-4, 4].
As with partial derivatives, we may treat one of the variables in f as constant
and think of the resulting function as a function of a single variable. Now we
investigate what happens if we integrate instead of differentiate.
a.
Choose a fixed value of x in the interior of [-3,3]. Let
What is the geometric meaning of the value of A(x) relative to the
surface defined by f. (Hint: Think about the trace determined by the fixed
value of x, and consider how A(x) is related to the image at left in
Figure 11.2.2.)
-4
-2 0
Y
4
A(z) = ª* f(x, y) dy.
-4
2
X
-1
Z 25
20
F15
10
+5
-4 -2 0
Y
2
X
Z
-3
25
20
15
10
5
Figure 11.2.2. Left: A cross section with fixed x. Right: A cross section with
fixed x and Ax.
X
b. For a fixed value of x, say x, what is the geometric meaning of
A(x) Ax? (Hint: Consider how A(x*)Aï is related to the image at right
in Figure 11.2.2.)
![b. For a fixed value of ï, say ¤‡, what is the geometric meaning of
A(x) Ax? (Hint: Consider how A(**)Ax is related to the image at right
in Figure 11.2.2.)
c. Since f is continuous on R, we can define the function A = A(x) at
every value of x in [−3, 3]. Now think about subdividing the x-interval
[-3,3] into m subintervals, and choosing a value * in each of those
subintervals. What will be the meaning of the sum ₁ A(x*) Ax?
Im
i=1
-3
d. Explain why ³ A(x) dx will determine the exact value of the volume
under the surface z = f(x, y) over the rectangle R.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F409330f8-4765-4598-8146-83752bc8a849%2F8f984369-035f-4684-b6ba-ad49b4c1cc25%2Ftt853i_processed.png&w=3840&q=75)
Transcribed Image Text:b. For a fixed value of ï, say ¤‡, what is the geometric meaning of
A(x) Ax? (Hint: Consider how A(**)Ax is related to the image at right
in Figure 11.2.2.)
c. Since f is continuous on R, we can define the function A = A(x) at
every value of x in [−3, 3]. Now think about subdividing the x-interval
[-3,3] into m subintervals, and choosing a value * in each of those
subintervals. What will be the meaning of the sum ₁ A(x*) Ax?
Im
i=1
-3
d. Explain why ³ A(x) dx will determine the exact value of the volume
under the surface z = f(x, y) over the rectangle R.
AI-Generated Solution
AI-generated content may present inaccurate or offensive content that does not represent bartleby’s views.
Unlock instant AI solutions
Tap the button
to generate a solution
Recommended textbooks for you
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage