Force on a Submerged Sheet In Exercises 1 and 2, the area of the top side of a piece of sheet metal is given. The sheet metal is submerged horizontally in 5 feet of water. Find the fluid force on the top side. 3 square feet
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P
Preparation for Calculus
P.1
Graphs and Models
P.2
Linear Models and Rates of Change
P.3
Functions and Their Graphs
P.4
Fitting Models to Data
1
Limits and Their Properties
1.1
A Preview of Calculus
1.2
Finding Limits Graphically and Numerically
1.3
Evaluating Limits Analytically
1.4
Continuity and One-Sided Limits
1.5
Infinite Limits
2
Differentiation
2.1
The Derivative and the Tangent Line Problem
2.2
Basic Differentiation Rules and Rates of Change
2.3
Product and Quotient Rules and Higher-Order Derivatives
2.4
The Chain Rule
2.5
Implicit Differentiation
2.6
Related Rates
3
Applications of Differentiation
3.1
Extrema on an Interval
3.2
Rolles Theorem and the Mean Value Theorem
3.3
Increasing and Decreasing Functions and the First Derivative Test
3.4
Concavity and the Second Derivative Test
3.5
Limits at Infinity
3.6
A Summary of Curve Sketching
3.7
Optimization Problems
3.8
Newtons Method
3.9
Differentials
4
Integration
4.1
Antiderivatives and Indefinite Integration
4.2
Area
4.3
Riemann Sums and Definite Integrals
4.4
The Fundamental Theorem of Calculus
4.5
Integration by Substitution
4.6
Numerical Integration
5
Logarithmic, Exponential, and Other Transcendental Functions
5.1
The Natural Logarithmic Function: Differentiation
5.2
The Natural Logarithmic Function: Integration
5.3
Inverse Functions
5.4
Exponential Functions: Differentiation and Integration
5.5
Bases Other Than e and Applications
5.6
Inverse Trigonometric Functions: Differentiation
5.7
Inverse Trigonometric Functions: Integration
5.8
Hyperbolic Functions
6
Differential Equations
6.1
Slope Fields and Eulers Method
6.2
Differential Equations: Growth and Decay
6.3
Separation of Variables and the Logistic Equation
6.4
First-Order Linear Differential Equations
7
Applications of Integration
7.1
Area of a Region Between Two Curves
7.2
Volume: The Disk Method
7.3
Volume: The Shell Method
7.4
Arc Length and Surfaces of Revolution
7.5
Work
7.6
Moments, Centers of Mass, and Centroids
7.7
Fluid Pressure and Fluid Force
8
Integration Techniques, LHpitals Rule, and Improper Integrals
8.1
Basic Integration Rules
8.2
Integration by Parts
8.3
Trigonometric Integrals
8.4
Trigonometric Substitution
8.5
Partial Fractions
8.6
Integration by Tables and Other Integration Techniques
8.7
Indeterminate Forms and LHpitals Rule
8.8
Improper Integrals
9
Infinite Series
9.1
Sequences
9.10
Taylor and Maclaurin Series
9.2
Series and Convergence
9.3
The Integral Test and p-Series
9.4
Comparisons of Series
9.5
Alternating Series
9.6
The Ratio and Root Tests
9.7
Taylor Polynomials and Approximations
9.8
Power Series
9.9
Representation of Functions by Power Series
10
Conics, Parametric Equations, and Polar Coordinates
10.1
Conics and Calculus
10.2
Plane Curves and Parametric Equations
10.3
Parametric Equations and Calculus
10.4
Polar Coordinates and Polar Graphs
10.5
Area and Arc Length in Polar Coordinates
10.6
Polar Equations of Conics and Keplers Laws
11
Vectors and the Geometry of Space
11.1
Vectors in the Plane
11.2
Space Coordinates and Vectors in Space
11.3
The Dot Product of Two Vectors
11.4
The Cross Product of Two Vectors in Space
11.5
Lines and Planes in Space
11.6
Surfaces in Space
11.7
Cylindrical and Spherical Coordinates
12
Vector-Valued Functions
12.1
Vector-Valued Functions
12.2
Differentiation and Integration of Vector-Valued Functions
12.3
Velocity and Acceleration
12.4
Tangent Vectors and Normal Vectors
12.5
Arc Length and Curvature
13
Functions of Several Variables
13.1
Introduction to Functions of Several Variables
13.10
Lagrange Multipliers
13.2
Limits and Continuity
13.3
Partial Derivatives
13.4
Differentials
13.5
Chain Rules for Functions of Several Variables
13.6
Directional Derivatives and Gradients
13.7
Tangent Planes and Normal Lines
13.8
Extrema of Functions of Two Variables
13.9
Applications of Extrema of Functions of Two Variables
14
Multiple Integration
14.1
Iterated Integrals and Area in the Plane
14.2
Double Integrals and Volume
14.3
Change of Variables: Polar Coordinates
14.4
Center of Mass and Moments of Inertia
14.5
Surface Area
14.6
Triple Integrals and Applications
14.7
Triple Integrals in Cylindrical and Spherical Coordinates
14.8
Change of Variables: Jacobians
15
Vector Analysis
15.1
Vector Fields
15.2
Line Integrals
15.3
Conservative Vector Fields and Independence of Path
15.4
Greens Theorem
15.5
Parametric Surfaces
15.6
Surface Integrals
15.7
Divergence Theorem
15.8
Stokess Theorem
Textbook Solutions for Calculus
Chapter 7.7 Problem 21
Question
Fluid Force on a Circular Plate A circular plate of radius feet is submerged vertically in a tank of fluid that weighs pounds per cubic foot. The center of the circle is feet below the surface of the fluid. Show that the fluid force on the surface of the plate is (Evaluate one integral by a geometric formula and the other by observing that the integrand is an odd function.)
Solution
The first step in solving 7.7 problem number 21 trying to solve the problem we have to refer to the textbook question: Fluid Force on a Circular Plate A circular plate of radius feet is submerged vertically in a tank of fluid that weighs pounds per cubic foot. The center of the circle is feet below the surface of the fluid. Show that the fluid force on the surface of the plate is (Evaluate one integral by a geometric formula and the other by observing that the integrand is an odd function.)
From the textbook chapter Fluid Pressure and Fluid Force you will find a few key concepts needed to solve this.
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full solution
Title
Calculus 8
Author
Ron Larson Robert P. Hostetler, Bruce H. Edwards
ISBN
9780618502981
Fluid Force on a Circular Plate A circular plate of radius
Chapter 7.7 textbook questions
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Chapter 7: Problem 1 Calculus 8
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Chapter 7: Problem 2 Calculus 8
Force on a Submerged Sheet In Exercises 1 and 2, the area of the top side of a piece of sheet metal is given. The sheet metal is submerged horizontally in 5 feet of water. Find the fluid force on the top side. 16 square feet
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Chapter 7: Problem 3 Calculus 8
Buoyant Force In Exercises 3 and 4, find the buoyant force of a rectangular solid of the given dimensions submerged in water so that the top side is parallel to the surface of the water. The buoyant force is the difference between the fluid forces on the top and bottom sides of the solid.
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Chapter 7: Problem 4 Calculus 8
Buoyant Force In Exercises 3 and 4, find the buoyant force of a rectangular solid of the given dimensions submerged in water so that the top side is parallel to the surface of the water. The buoyant force is the difference between the fluid forces on the top and bottom sides of the solid.
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Chapter 7: Problem 5 Calculus 8
Fluid Force on a Tank Wall In Exercises 510, find the fluid force on the vertical side of the tank, where the dimensions are given in feet. Assume that the tank is full of water. Rectangle
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Chapter 7: Problem 6 Calculus 8
Fluid Force on a Tank Wall In Exercises 510, find the fluid force on the vertical side of the tank, where the dimensions are given in feet. Assume that the tank is full of water. Triangle
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Chapter 7: Problem 7 Calculus 8
Fluid Force on a Tank Wall In Exercises 510, find the fluid force on the vertical side of the tank, where the dimensions are given in feet. Assume that the tank is full of water. Trapezoid
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Chapter 7: Problem 8 Calculus 8
Fluid Force on a Tank Wall In Exercises 510, find the fluid force on the vertical side of the tank, where the dimensions are given in feet. Assume that the tank is full of water. Semicircle
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Chapter 7: Problem 9 Calculus 8
Fluid Force on a Tank Wall In Exercises 510, find the fluid force on the vertical side of the tank, where the dimensions are given in feet. Assume that the tank is full of water. Parabola
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Chapter 7: Problem 10 Calculus 8
Fluid Force on a Tank Wall In Exercises 510, find the fluid force on the vertical side of the tank, where the dimensions are given in feet. Assume that the tank is full of water. Semiellipse
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Chapter 7: Problem 11 Calculus 8
Fluid Force of Water In Exercises 1114, find the fluid force on the vertical plate submerged in water, where the dimensions are given in meters and the weight-density of water is 9800 newtons per cubic meter. Square
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Chapter 7: Problem 12 Calculus 8
Fluid Force of Water In Exercises 1114, find the fluid force on the vertical plate submerged in water, where the dimensions are given in meters and the weight-density of water is 9800 newtons per cubic meter. Square
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Chapter 7: Problem 13 Calculus 8
Fluid Force of Water In Exercises 1114, find the fluid force on the vertical plate submerged in water, where the dimensions are given in meters and the weight-density of water is 9800 newtons per cubic meter. Triangle
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Chapter 7: Problem 14 Calculus 8
Fluid Force of Water In Exercises 1114, find the fluid force on the vertical plate submerged in water, where the dimensions are given in meters and the weight-density of water is 9800 newtons per cubic meter. Rectangle
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Chapter 7: Problem 15 Calculus 8
Force on a Concrete Form In Exercises 1518, the figure is the vertical side of a form for poured concrete that weighs 140.7 pounds per cubic foot. Determine the force on this part of the concrete form. Rectangle
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Chapter 7: Problem 16 Calculus 8
Force on a Concrete Form In Exercises 1518, the figure is the vertical side of a form for poured concrete that weighs 140.7 pounds per cubic foot. Determine the force on this part of the concrete form. Semiellipse
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Chapter 7: Problem 17 Calculus 8
Force on a Concrete Form In Exercises 1518, the figure is the vertical side of a form for poured concrete that weighs 140.7 pounds per cubic foot. Determine the force on this part of the concrete form. Rectangle
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Chapter 7: Problem 18 Calculus 8
Force on a Concrete Form In Exercises 1518, the figure is the vertical side of a form for poured concrete that weighs 140.7 pounds per cubic foot. Determine the force on this part of the concrete form. Triangle
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Chapter 7: Problem 19 Calculus 8
Fluid Force of Gasoline A cylindrical gasoline tank is placed so that the axis of the cylinder is horizontal. Find the fluid force on a circular end of the tank if the tank is half full, assuming that the diameter is 3 feet and the gasoline weighs 42 pounds per cubic foot.
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Chapter 7: Problem 20 Calculus 8
Fluid Force of Gasoline Repeat Exercise 19 for a tank that is full. (Evaluate one integral by a geometric formula and the other by observing that the integrand is an odd function.)
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Chapter 7: Problem 21 Calculus 8
Fluid Force on a Circular Plate A circular plate of radius feet is submerged vertically in a tank of fluid that weighs pounds per cubic foot. The center of the circle is feet below the surface of the fluid. Show that the fluid force on the surface of the plate is (Evaluate one integral by a geometric formula and the other by observing that the integrand is an odd function.)
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Chapter 7: Problem 22 Calculus 8
Fluid Force on a Circular Plate Use the result of Exercise 21 to find the fluid force on the circular plate shown in each figure. Assume the plates are in the wall of a tank filled with water and the measurements are given in feet. (a) (b)
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Chapter 7: Problem 23 Calculus 8
Fluid Force on a Rectangular Plate A rectangular plate of height feet and base feet is submerged vertically in a tank of fluid that weighs pounds per cubic foot. The center is feet below the surface of the fluid, where Show that the fluid force on the surface of the plate is
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Chapter 7: Problem 24 Calculus 8
Fluid Force on a Rectangular Plate Use the result of Exercise 23 to find the fluid force on the rectangular plate shown in each figure. Assume the plates are in the wall of a tank filled with water and the measurements are given in feet. (a) (b)
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Chapter 7: Problem 25 Calculus 8
Submarine Porthole A porthole on a vertical side of a submarine (submerged in seawater) is 1 square foot. Find the fluid force on the porthole, assuming that the center of the square is 15 feet below the surface.
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Chapter 7: Problem 26 Calculus 8
Submarine Porthole Repeat Exercise 25 for a circular porthole that has a diameter of 1 foot. The center is 15 feet below the surface.
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Chapter 7: Problem 27 Calculus 8
Modeling Data The vertical stern of a boat with a superimposed coordinate system is shown in the figure. The table shows the width of the stern at indicated values of Find the fluid force against the stern if the measurements are given in feet.
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Chapter 7: Problem 28 Calculus 8
Irrigation Canal Gate The vertical cross section of an irrigation canal is modeled by where is measured in feet and corresponds to the center of the canal. Use the integration capabilities of a graphing utility to approximate the fluid force against a vertical gate used to stop the flow of water if the water is 3 feet deep.
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Chapter 7: Problem 29 Calculus 8
In Exercises 29 and 30, use the integration capabilities of a graphing utility to approximate the fluid force on the vertical plate bounded by the -axis and the top half of the graph of the equation. Assume that the base of the plate is 12 feet beneath the surface of the water 23 y23 423 x
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Chapter 7: Problem 30 Calculus 8
In Exercises 29 and 30, use the integration capabilities of a graphing utility to approximate the fluid force on the vertical plate bounded by the -axis and the top half of the graph of the equation. Assume that the base of the plate is 12 feet beneath the surface of the water x2 28 y2 16 x 1
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Chapter 7: Problem 31 Calculus 8
Think About It (a) Approximate the depth of the water in the tank in Exercise 5 if the fluid force is one-half as great as when the tank is full. (b) Explain why the answer in part (a) is not 3 2.
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Chapter 7: Problem 33 Calculus 8
Define fluid force against a submerged vertical plane region.
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