Let R be the region in the first quadrant that is enclosed by y = f(z) (or z = g(y)) and y = z as shown below given that a = 1, b = 5, c=1, and d = 5. f y=x C- a Part a: Which of the following will give the PERIMETER of the region R: of* √ ₁ + (1)³² dz + + S* √² + (ƒ^(z))}³ dz °f* √ ¹ + (ƒ¨(z)) ² dz 0² √ ₁ + (ƒ²(z) + 1)² dz 0 [° √¹ + (1)³ dz + [° √ ¹ + (ƒ(z))³ dz Part b: If the region R is the base of a solid with cross-sections perpendicular to the z-axis (and to the zy- plane) that are triangles with height equal to its base. Setup the definite integral to find the volume of this solid. Volume = -1 dz Part c: If the region R is the base of a solid with cross-sections perpendicular to the y-axis (and to the zy- plane) that are circles. Setup the definite integral to find the volume of this solid. Volume = -1 dy y = f(x) or x = g(y)
Let R be the region in the first quadrant that is enclosed by y = f(z) (or z = g(y)) and y = z as shown below given that a = 1, b = 5, c=1, and d = 5. f y=x C- a Part a: Which of the following will give the PERIMETER of the region R: of* √ ₁ + (1)³² dz + + S* √² + (ƒ^(z))}³ dz °f* √ ¹ + (ƒ¨(z)) ² dz 0² √ ₁ + (ƒ²(z) + 1)² dz 0 [° √¹ + (1)³ dz + [° √ ¹ + (ƒ(z))³ dz Part b: If the region R is the base of a solid with cross-sections perpendicular to the z-axis (and to the zy- plane) that are triangles with height equal to its base. Setup the definite integral to find the volume of this solid. Volume = -1 dz Part c: If the region R is the base of a solid with cross-sections perpendicular to the y-axis (and to the zy- plane) that are circles. Setup the definite integral to find the volume of this solid. Volume = -1 dy y = f(x) or x = g(y)
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.3: Lines
Problem 10E
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![Let \( R \) be the region in the first quadrant that is enclosed by \( y = f(x) \) (or \( x = g(y) \)) and \( y = x \) as shown below given that \( a = 1 \), \( b = 5 \), \( c = 1 \), and \( d = 5 \).
[There is a diagram illustrating the region \( R \) in the first quadrant, bounded by the blue curve \( y = f(x) \) or \( x = g(y) \), the red line \( y = x \), and the vertical lines at \( x = a = 1 \) and \( x = b = 5 \). The vertical axis is marked as \( d \) and the horizontal axis is marked as \( c \). The region resembles a slice bound by aforementioned curves and lines.]
**Part a: Which of the following will give the PERIMETER of the region \( R \):**
\[
\begin{array}{l}
1. \quad \int_{1}^{5} \sqrt{1 + (1)^{2}} \, dx + \int_{1}^{5} \sqrt{1 + (f'(x))^{2}} \, dx \\
2. \quad \int_{1}^{5} \sqrt{1 + (f'(x))^{2}} \, dx \\
3. \quad \int_{1}^{5} \sqrt{1 + (f'(x) + 1)^{2}} \, dx \\
4. \quad \int_{1}^{5} \sqrt{1 + (1)^{2}} \, dx + \int_{1}^{5} \sqrt{1 + (f(x))^{2}} \, dx
\end{array}
\]
**Part b: If the region \( R \) is the base of a solid with cross-sections perpendicular to the x-axis (and to the xy-plane) that are triangles with height equal to its base:**
Setup the definite integral to find the volume of this solid.
\[
\text{Volume} = \int_{1}^{5} \left( \text{(to be filled)} \right) \, dx
\]
**Part c: If the region \( R \) is the base of a solid with cross-sections perpendicular to the y-axis (and to](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7f31f885-3acd-4362-8d52-f6c21adcaba6%2Fe5b3715a-7d66-4bd8-9c5c-a7113e17acf5%2F6bogzh9_processed.png&w=3840&q=75)
Transcribed Image Text:Let \( R \) be the region in the first quadrant that is enclosed by \( y = f(x) \) (or \( x = g(y) \)) and \( y = x \) as shown below given that \( a = 1 \), \( b = 5 \), \( c = 1 \), and \( d = 5 \).
[There is a diagram illustrating the region \( R \) in the first quadrant, bounded by the blue curve \( y = f(x) \) or \( x = g(y) \), the red line \( y = x \), and the vertical lines at \( x = a = 1 \) and \( x = b = 5 \). The vertical axis is marked as \( d \) and the horizontal axis is marked as \( c \). The region resembles a slice bound by aforementioned curves and lines.]
**Part a: Which of the following will give the PERIMETER of the region \( R \):**
\[
\begin{array}{l}
1. \quad \int_{1}^{5} \sqrt{1 + (1)^{2}} \, dx + \int_{1}^{5} \sqrt{1 + (f'(x))^{2}} \, dx \\
2. \quad \int_{1}^{5} \sqrt{1 + (f'(x))^{2}} \, dx \\
3. \quad \int_{1}^{5} \sqrt{1 + (f'(x) + 1)^{2}} \, dx \\
4. \quad \int_{1}^{5} \sqrt{1 + (1)^{2}} \, dx + \int_{1}^{5} \sqrt{1 + (f(x))^{2}} \, dx
\end{array}
\]
**Part b: If the region \( R \) is the base of a solid with cross-sections perpendicular to the x-axis (and to the xy-plane) that are triangles with height equal to its base:**
Setup the definite integral to find the volume of this solid.
\[
\text{Volume} = \int_{1}^{5} \left( \text{(to be filled)} \right) \, dx
\]
**Part c: If the region \( R \) is the base of a solid with cross-sections perpendicular to the y-axis (and to
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