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Approximating Arc Length or Surface Area In Exercises 65-68, write the definite integral for finding the indicated arc length or surface area. Then use the integration capabilities of a graphing utility to approximate the arc length or surface area. (You will learn how to evaluate this type of integral in Section 8.8.) Length of Pursuit A fleeing object leaves the origin and moves up the y -axis (see figure). At the same time, a pursuer leaves the point (1,0) and always moves toward the fleeing object. The pursuer’s speed is twice that of the fleeing object. The equation of the path is modeled by y = 1 3 ( x 3 / 2 − 3 x 1 / 2 + 2 ) . How far has the fleeing object traveled when it is caught? Show that the pursuer has traveled twice as far.

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Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781337552516

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Chapter
Section
BuyFindarrow_forward

Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781337552516
Chapter 7.4, Problem 65E
Textbook Problem
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Approximating Arc Length or Surface Area In Exercises 65-68, write the definite integral for finding the indicated arc length or surface area. Then use the integration capabilities of a graphing utility to approximate the arc length or surface area. (You will learn how to evaluate this type of integral in Section 8.8.)

Length of Pursuit A fleeing object leaves the origin and moves up the y-axis (see figure). At the same time, a pursuer leaves the point (1,0) and always moves toward the fleeing object. The pursuer’s speed is twice that of the fleeing object. The equation of the path is modeled by

y = 1 3 ( x 3 / 2 3 x 1 / 2 + 2 ) .

How far has the fleeing object traveled when it is caught? Show that the pursuer has traveled twice as far.

Chapter 7.4, Problem 65E, Approximating Arc Length or Surface Area In Exercises 65-68, write the definite integral for finding

To determine

To calculate: The length of the arc for the function y=13(x323x12+2) and that the distance travelled by pursuer is twice than the object.

Explanation of Solution

Given:

The equation of the path is y=13(x323x12+2). The movement is as shown in the figure:

Formula used:

The length of the arc can be found by using integration, from the following formula

L=ab1+f(x)2dx

Where d is the length of the arc and f(x) represents the equation of the function representing graph.

Calculation:

The function is y=13(x323x12+2)

When the object starts travelling, x=0

Thus, the total distance travelled by the object can be calculated by putting x=0 in y

As follows,

y=13((0)323(0)12+2)=13(0+0+2)=23

Thus, the object travelled 23units when it is caught.

To calculate the length of the arc, the derivative is calculated;

y=13ddx(x323x12+2)=13(32x32232x122+0)=x1221x12

Taking LCM,

y=12(x1x12)

Squaring and adding ‘1’

1+y2=1+(12(x1x12))2=1+(x1)24x=4x

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Chapter 7 Solutions

Calculus: Early Transcendental Functions
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The graphs of the functions f1,...Ch. 7.4 - Think About It The figure shows the graphs of the...Ch. 7.4 - Verifying a Formula (a) Given a circular sector...Ch. 7.4 - Lateral Surface Area of a Cone A right circular...Ch. 7.4 - Lateral Surface Area of a Cone A right circular...Ch. 7.4 - Using a Sphere Find the area of the segment of a...Ch. 7.4 - Using a Sphere Find the area of the segment of a...Ch. 7.4 - Modeling Data The circumference C (in inches) of a...Ch. 7.4 - Modeling Data Property bounded by two...Ch. 7.4 - Volume and Surface Area Let R be the region...Ch. 7.4 - Think About It Consider the equation x29+y24=1....Ch. 7.4 - Approximating Arc Length or Surface Area In...Ch. 7.4 - Approximating Arc Length or Surface Area In...Ch. 7.4 - Approximating Arc Length or Surface Area In...Ch. 7.4 - Approximating Arc Length or Surface Area In...Ch. 7.4 - Suspension Bridge A cable for a suspension bridge...Ch. 7.4 - Suspension Bridge The Humber Bridge, located in...Ch. 7.4 - Arc Length and Area Let C be the curve given by...Ch. 7.4 - PUTNAM EXAM CHALLENGE Find the length of the curve...Ch. 7.5 - CONCEPT CHECK Work How do you know when work is...Ch. 7.5 - CONCEPT CHECK Comparing Methods Describe the...Ch. 7.5 - CONCEPT CHECK Hooke's Law Describe Hookes Law in...Ch. 7.5 - CONCEPT CHECK Work What are two ways to write the...Ch. 7.5 - Constant Force In Exercises 5-8, determine the...Ch. 7.5 - Constant Force In Exercises 5-8, determine the...Ch. 7.5 - Constant Force In Exercises 5-8, determine the...Ch. 7.5 - Constant Force In Exercises 5-8, determine the...Ch. 7.5 - Hooke's Law In Exercises 9-14, use Hookes Law to...Ch. 7.5 - Hooke's Law In Exercises 9-14, use Hookes Law to...Ch. 7.5 - Hooke's Law In Exercises 9-14, use Hookes Law to...Ch. 7.5 - Hooke's Law In Exercises 9-14, use Hookes Law to...Ch. 7.5 - Hooke's Law In Exercises 9-14, use Hookes Law to...Ch. 7.5 - Hooke's Law In Exercises 9-14, use Hookes Law to...Ch. 7.5 - Propulsion Neglecting air resistance and the...Ch. 7.5 - Propulsion Use the information in Exercise 15 to...Ch. 7.5 - Propulsion Neglecting air resistance and the...Ch. 7.5 - Propulsion A lunar module weighs 12 metric tons on...Ch. 7.5 - Pumping Water A rectangular tank with a base 4...Ch. 7.5 - Think About It Explain why the answer in part (b)...Ch. 7.5 - Pumping Water A cylindrical water tank 4 meters...Ch. 7.5 - Pumping Water Suppose the tank in Exercise 21 is...Ch. 7.5 - Pumping Water An open tank has the shape of a...Ch. 7.5 - Pumping Water Water is pumped in through the...Ch. 7.5 - Pumping Water A hemispherical tank of radius 6...Ch. 7.5 - Pumping Diesel Fuel The fuel tank on a truck has...Ch. 7.5 - Pumping Gasoline In Exercises 27 and 28, find the...Ch. 7.5 - Pumping Gasoline In Exercises 27 and 28, find the...Ch. 7.5 - Winding a Chain In Exercises 29-32, consider a...Ch. 7.5 - Winding a Chain In Exercises 29-32, consider a...Ch. 7.5 - Winding a Chain In Exercises 29-32, consider a...Ch. 7.5 - Winding a Chain In Exercises 29-32, consider a...Ch. 7.5 - Lifting a Chain In Exercises 33 and 34, consider a...Ch. 7.5 - Lifting a Chain In Exercises 33 and 34, consider a...Ch. 7.5 - EXPLORING CONCEPTS Think AboutIt Does it take any...Ch. 7.5 - EXPLORING CONCEPTS, Think About It In Example 1,...Ch. 7.5 - Newton's Law of Universal Gravitation Consider two...Ch. 7.5 - Conjecture Use Newtons Law of Universal...Ch. 7.5 - Electric Force Two electrons repel each other with...Ch. 7.5 - HOW DO YOU SEE IT? The graphs show the force Fi...Ch. 7.5 - Ordering Forces Verify your answer to Exercise 40...Ch. 7.5 - Comparing Work Order the following from least to...Ch. 7.5 - Boyle's Law In Exercises 43 and 44, find the work...Ch. 7.5 - Boyle's Law In Exercises 43 and 44, find the work...Ch. 7.5 - Hydraulic Press In Exercises 45-48, use the...Ch. 7.5 - Hydraulic Press In Exercises 45-48, use the...Ch. 7.5 - Hydraulic Press In Exercises 45-48, use the...Ch. 7.5 - Hydraulic Press In Exercises 45-48, use the...Ch. 7.5 - Modeling Data The hydraulic cylinder on a...Ch. 7.6 - CONCEPT CHECK Mass and Weight How are mass and...Ch. 7.6 - CONCEPT CHECK Moment The equation for the moment...Ch. 7.6 - CONCEPT CHECK Planar Lamina What is a planar...Ch. 7.6 - CONCEPT CHECK Theorem of Pappus Explain why the...Ch. 7.6 - Center of Mass of a Linear System In Exercises...Ch. 7.6 - Center of Mass of a Linear System In Exercises...Ch. 7.6 - Center of Mass of a Linear System In Exercises...Ch. 7.6 - Center of Mass of a Linear System In Exercises...Ch. 7.6 - Equilibrium of a Linear System In Exercises 9 and...Ch. 7.6 - Equilibrium of a Linear System In Exercises 9 and...Ch. 7.6 - Center of Mass of a Two-Dimensional System In...Ch. 7.6 - Center of Mass of a Two-Dimensional System In...Ch. 7.6 - Center of Mass of a Two-Dimensional System In...Ch. 7.6 - Center of Mass of a Two-Dimensional System In...Ch. 7.6 - Center of Mass of a Planar Lamina In Exercises...Ch. 7.6 - Center of Mass of a Planar Lamina In Exercises...Ch. 7.6 - Center of Mass of a Planar Lamina In Exercises...Ch. 7.6 - Center of Mass of a Planar Lamina In Exercises...Ch. 7.6 - Center of Mass of a Planar Lamina In Exercises...Ch. 7.6 - Center of Mass of a Planar Lamina In Exercises...Ch. 7.6 - Center of Mass of a Planar Lamina In Exercises...Ch. 7.6 - Center of Mass of a Planar Lamina In Exercises...Ch. 7.6 - Center of Mass of a Planar Lamina In Exercises...Ch. 7.6 - Center of Mass of a Planar Lamina In Exercises...Ch. 7.6 - Center of Mass of a Planar Lamina In Exercises...Ch. 7.6 - Center of Mass of a Planar Lamina In Exercises...Ch. 7.6 - Center of Mass of a Planar Lamina In Exercises...Ch. 7.6 - Center of Mass of a Planar Lamina In Exercises...Ch. 7.6 - Approximating a Centroid Using Technology In...Ch. 7.6 - Approximating a Centroid Using Technology In...Ch. 7.6 - Finding the Center of Mass In Exercises 31-34,...Ch. 7.6 - Finding the Center of Mass In Exercises 31-34,...Ch. 7.6 - Finding the Center of Mass In Exercises 31-34,...Ch. 7.6 - Finding the Center of Mass In Exercises 31-34,...Ch. 7.6 - Finding the Center of Mass Find the center of mass...Ch. 7.6 - Finding the Center of Mass Find the center of mass...Ch. 7.6 - Finding Volume by the Theorem of Pappus In...Ch. 7.6 - Finding Volume by the Theorem of Pappus In...Ch. 7.6 - Finding Volume by the Theorem of Pappus In...Ch. 7.6 - Finding Volume by the Theorem of Pappus In...Ch. 7.6 - EXPLORING CONCEPTS Center of Mass What happens to...Ch. 7.6 - EXPLORING CONCEPTS Centroid Explain why the...Ch. 7.6 - EXPLORING CONCEPTS Center of Mass Use rectangles...Ch. 7.6 - HOW DO YOU SEE IT? The centroid of the plane...Ch. 7.6 - Centroid of a Common Region In Exercises 45-50,...Ch. 7.6 - Centroid of a Common Region In Exercises 45-50,...Ch. 7.6 - Centroid of a Common Region In Exercises 45-50,...Ch. 7.6 - Centroid of a Common Region In Exercises 45-50,...Ch. 7.6 - Centroid of a Common Region In Exercises 45-50,...Ch. 7.6 - Centroid of a Common Region In Exercises 45-50,...Ch. 7.6 - Graphical Reasoning Consider the region bounded by...Ch. 7.6 - Graphical and Numerical Reasoning Consider the...Ch. 7.6 - Modeling Data The manufacturer of glass for a...Ch. 7.6 - Modeling Data The manufacturer of a boat needs...Ch. 7.6 - Second Theorem of Pappus In Exercises 55 and 56,...Ch. 7.6 - Second Theorem of Pappus In Exercises 55 and 56,...Ch. 7.6 - Finding a Centroid Let n1 be constant, and...Ch. 7.6 - Finding a Centroid Consider the functions...Ch. 7.6 - PUTNAM EXAM CHALLENGE Let V be the region in the...Ch. 7.7 - CONCEPT CHECK Fluid Pressure Describe fluid...Ch. 7.7 - CONCEPT CHECK Fluid Pressure Does fluid pressure...Ch. 7.7 - Force on a Submerged Sheet In Exercises 3-6, the...Ch. 7.7 - Force on a Submerged Sheet In Exercises 3-6, the...Ch. 7.7 - Force on a Submerged Sheet In Exercises 3-6, the...Ch. 7.7 - Force on a Submerged Sheet In Exercises 3-6, the...Ch. 7.7 - Force on a Submerged Sheet In Exercises 7 and 8,...Ch. 7.7 - Force on a Submerged Sheet In Exercises 7 and 8,...Ch. 7.7 - Fluid Force on a Tank Wall In Exercises 9-14, find...Ch. 7.7 - Fluid Force on a Tank Wall In Exercises 9-14, find...Ch. 7.7 - Fluid Force on a Tank Wall In Exercises 9-14, find...Ch. 7.7 - Fluid Force on a Tank Wall In Exercises 9-14, find...Ch. 7.7 - Fluid Force on a Tank Wall In Exercises 9-14, find...Ch. 7.7 - Fluid Force on a Tank Wall In Exercises 9-14, find...Ch. 7.7 - Fluid Force of Water In Exercises 15-18, Find the...Ch. 7.7 - Fluid Force of Water In Exercises 15-18, Find the...Ch. 7.7 - Fluid Force of Water In Exercises 15-18, Find the...Ch. 7.7 - Fluid Force of Water In Exercises 15-18, Find the...Ch. 7.7 - Force on a Concrete Form In Exercises 19-22, the...Ch. 7.7 - Force on a Concrete Form In Exercises 19-22, the...Ch. 7.7 - Force on a Concrete Form In Exercises 19-22, the...Ch. 7.7 - Force on a Concrete Form In Exercises 19-22, the...Ch. 7.7 - Fluid Force of Gasoline A cylindrical gasoline...Ch. 7.7 - Fluid Force of Gasoline Repeat Exercise 23 for a...Ch. 7.7 - EXPLORING CONCEPTS Fluid Pressure Explain why...Ch. 7.7 - EXPLORING CONCEPTS Buoyant Force Buoyant force is...Ch. 7.7 - EXPLORING CONCEPTS Think About It Approximate the...Ch. 7.7 - HOW DO YOU SEE IT? Two identical semicircular...Ch. 7.7 - Fluid Force on a Circular Plate A circular plate...Ch. 7.7 - Fluid Force on a Circular Plate Use the result of...Ch. 7.7 - Fluid Force on a Rectangular Plate A rectangular...Ch. 7.7 - Fluid Force on a Rectangular Plate Use the result...Ch. 7.7 - Submarine Porthole A square porthole on a vertical...Ch. 7.7 - Submarine Porthole Repeat Exercise 33 for a...Ch. 7.7 - Modeling Data The vertical stem of a boat...Ch. 7.7 - Irrigation Canal Gate The vertical cross section...Ch. 7 - Finding the Area of a Region In Exercises 1-10,...Ch. 7 - Finding the Area of a Region In Exercises 1-10,...Ch. 7 - Finding the Area of a Region In Exercises 1-10,...Ch. 7 - Finding the Area of a Region In Exercises 1-10,...Ch. 7 - Finding the Area of a Region In Exercises 1-10,...Ch. 7 - Finding the Area of a Region In Exercises 1-10,...Ch. 7 - Finding the Area of a Region In Exercises 1-10,...Ch. 7 - Finding the Area of a Region In Exercises 1-10,...Ch. 7 - Finding the Area of a Region In Exercises 1-10,...Ch. 7 - Finding the Area of a Region In Exercises 1-10,...Ch. 7 - Finding the Area of a Region In Exercises 11-14,...Ch. 7 - Finding the Area of a Region In Exercises 11-14,...Ch. 7 - Finding the Area of a Region In Exercises 11-14,...Ch. 7 - Finding the Area of a Region In Exercises 11-14,...Ch. 7 - Integration as an Accumulation Process In...Ch. 7 - Integration as an Accumulation Process In...Ch. 7 - Revenue In Exercises 17 and 18, two models R1, and...Ch. 7 - Revenue In Exercises 17 and 18, two models R1,...Ch. 7 - Finding the Volume of a Solid In Exercises 19 and...Ch. 7 - Finding the Volume of a Solid In Exercises 19 and...Ch. 7 - Finding the Volume of a Solid In Exercises 21 and...Ch. 7 - Finding the Volume of a Solid In Exercises 21 and...Ch. 7 - Finding the Volume of a Solid In Exercises 23 and...Ch. 7 - Finding the Volume of a Solid In Exercises 23 and...Ch. 7 - Gasoline Tank A gasoline tank is an oblate...Ch. 7 - Using Cross Sections Find the volume of the solid...Ch. 7 - Finding Arc Length In Exercises 27 and 28, find...Ch. 7 - Finding Arc Length In Exercises 27 and 28, find...Ch. 7 - Finding the Area of a Surface of Revolution In...Ch. 7 - Finding the Area of a Surface of Revolution In...Ch. 7 - Finding the Area of a Surface of Revolution In...Ch. 7 - Finding the Area of a Surface of Revolution In...Ch. 7 - Hooke's Law A force of 5 pounds stretches a spring...Ch. 7 - Hooke's Law A force of 50 pounds stretches a...Ch. 7 - Propulsion Neglecting air resistance and the...Ch. 7 - Pumping Water A water well has an 8-inch diameter...Ch. 7 - Winding a Chain A chain 10 feet long weighs 4...Ch. 7 - Winding a Cable A 200-foot cable weighing 5 pounds...Ch. 7 - Boyle's Law A quantity of gas with an initial...Ch. 7 - Boyle's Law A quantity of gas with an initial...Ch. 7 - Center of Mass of a Linear System Find the center...Ch. 7 - Center of Mass of a Two-Dimensional System Find...Ch. 7 - Center of Mass of a Planar Lamina In Exercises 43...Ch. 7 - Center of Mass of a Planar Lamina In Exercises 43...Ch. 7 - Finding the Center of Mass Introduce an...Ch. 7 - Finding Volume Use the Theorem of Pappus to find...Ch. 7 - Force on a Submerged Sheet In Exercises 47 and 48,...Ch. 7 - Force on a Submerged Sheet In Exercises 47 and 48,...Ch. 7 - Fluid Force of Seawater Find the fluid force on...Ch. 7 - Force on a Concrete Form The vertical side of a...Ch. 7 - Submarine Porthole A circular porthole on a...Ch. 7 - Finding a Limit Let R be the area of the region in...Ch. 7 - Center of Mass of a Lamina Let L. be the lamina of...Ch. 7 - Dividing a Region Let R be the region bounded by...Ch. 7 - Surface Area Graph the curve 8y2=x2(1x2). Use a...Ch. 7 - Centroid A blade on an industrial fan has the...Ch. 7 - Volume A hole is cut through the center of a...Ch. 7 - Volume A rectangle R of length l and width w is...Ch. 7 - Comparing Areas of Regions (a) The tangent line to...Ch. 7 - Using Arc Length The graph of y=f(x) passes...Ch. 7 - Using a Function Let f be rectifiable on the...Ch. 7 - Archimedes' Principle Archimedes Principle states...Ch. 7 - Finding a Centroid Sketch the region bounded on...Ch. 7 - Finding a Centroid Sketch the region bounded on...Ch. 7 - Work Find the work done by each force F.Ch. 7 - Consumer and Producer Surplus In Exercises 15 and...Ch. 7 - Consumer and Producer Surplus In Exercises 15 and...Ch. 7 - Fluid Force A swimming pool is 20 feet wide, 40...

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