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Calculus

10th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781285057095

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Section
BuyFindarrow_forward

Calculus

10th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781285057095
Chapter 7.1, Problem 55E
Textbook Problem
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Finding the Area of a Figure In Exercises 55–58, use integration to find the area of the figure having the given vertices.

(2, –3), (4, 6), (6, 1)

To determine

To calculate: The area of the region joined by the vertices (2,3),(4,6) and (6,1) with help of integration.

Explanation of Solution

Given:

The vertices are (2,3),(4,6) and (6,1).

Formula used:

Equation of line in two-point form is,

(yy1)=y2y1x2x1(xx1) where (x1,y1)and (x2,y2) are two points through which the line passes.

The integration formula,

xn=xn+1n+1+C

Area under between two curves f(x),g(x), where x varies from x1 to x2,

A=x1x2[f(x)g(x)]dx

Calculation:

Consider the 3 vertices, (2,3),(4,6) and (6,1).

First take the vertices (2,3)and(4,6).

Compute the equation of line passing through the two points.

Recall that equation of line in two-point form is (yy1)=y2y1x2x1(xx1), where

(x1,y1)and (x2,y2) are two points through which the line passes.

Substitute (2,3)and(4,6) for (x1,y1)and (x2,y2),

(y(3))=6(3)42(x2)(y+3)=6+32(x2)(y+3)=92(x2)2(y+3)=9(x2)

Solve ahead as,

2y+6=9x182y+6+18=9x2y+24=9xx=2y+249

The equation of the line joining the points (2,3)and(4,6) is x=2y+249.

Secondly take the vertices (4,6)and(6,1).

Compute the equation of line passing through the two points.

Recall that equation of line in two-point form is (yy1)=y2y1x2x1(xx1), where

(x1,y1)and (x2,y2) are two points through which the line passes.

Substitute (4,6)and(6,1) for (x1,y1)and (x2,y2),

(y6)=1664(x4)(y6)=52(x4)2(y6)=5(x4)2y12=5x+20

Solve ahead as,

5x20=2y+125x=2y+12+205x=2y+32x=2y+325

The equation of the line joining the points (4,6)and(6,1) is x=2y+325.

Thirdly take the vertices (2,3)and(6,1).

Compute the equation of line passing through the two points.

Recall that equation of line in two-point form is (yy1)=y2y1x2x1(xx1), where

(x1,y1)and (x2,y2) are two points through which the line passes.

Substitute (2,3)and(6,1) for (x1,y1)and (x2,y2),

(y(3))=1(3)62(x2)(y+3)=1+34(x2)(y+3)=44(x2)y+3=x2

Solve ahead as,

x=y+5

The equation of the line joining the points (2,3)and(6,1) is x=y+5

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Chapter 7 Solutions

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The graphs show the force Fi...Ch. 7.5 - Electric Force Two electrons repel each other with...Ch. 7.5 - Ordering Forces Verify your answer to Exercise 40...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 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 14,...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 - Graphical Reasoning (a) Translate each point mass...Ch. 7.6 - Conjecture Use the result of Exercise 5 to make a...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 In Exercises 2730, use a...Ch. 7.6 - Approximating a Centroid In Exercises 2730, use a...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 - Center of Mass Let the point masses m1, m2...,...Ch. 7.6 - Planar Lamina What is a planar lamina? Describe...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 - PUTNAM EXAM CHALLENGE Let V be the region in the...Ch. 7.7 - Force on a Submerged Sheet In Exercises 3-6, the...Ch. 7.7 - Force on a Submerged Sheet In Exercises 36, the...Ch. 7.7 - Force on a Submerged Sheet In Exercises 36, the...Ch. 7.7 - Force on a Submerged Sheet In Exercises 36, the...Ch. 7.7 - Buoyant Force In Exercises 5 and 6, find the...Ch. 7.7 - Buoyant Force In Exercises 5 and 6, find the...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 914, find...Ch. 7.7 - Fluid Force on a Tank Wall In Exercises 914, find...Ch. 7.7 - Fluid Force on a Tank Wall In Exercises 914, find...Ch. 7.7 - Fluid Force on a Tank Wall In Exercises 914, find...Ch. 7.7 - Fluid Force on a Tank Wall In Exercises 914, find...Ch. 7.7 - Fluid Force of a Water In Exercises 15-18, find...Ch. 7.7 - Fluid Force of Water In Exercises 15-18, find the...Ch. 7.7 - Fluid Force of Water In Exercises 1518, find the...Ch. 7.7 - Fluid Force of Water In Exercises 1518, find the...Ch. 7.7 - Force on a Concrete Form In Exercises 19-22, the...Ch. 7.7 - Force on a Concrete formIn Exercises 1922, 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 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 Pothole Repeat Exercise 33 for a...Ch. 7.7 - Modeling Data The vertical stem of a boat with a...Ch. 7.7 - Irrigation Canal Gate The vertical cross section...Ch. 7 - Finding the Area of a RegionIn Exercises 110,...Ch. 7 - Finding the Area of a RegionIn Exercises 1-10,...Ch. 7 - Finding the Area of a RegionIn Exercises 1-10,...Ch. 7 - Finding the Area of a RegionIn Exercises 1-10,...Ch. 7 - Finding the Area of a RegionIn Exercises 1-10,...Ch. 7 - Finding the Area of a RegionIn Exercises 1-10,...Ch. 7 - Finding the Area of a RegionIn Exercises 1-10,...Ch. 7 - Finding the Area of a RegionIn Exercises 1-10,...Ch. 7 - Finding the Area of a Region In Exercises 1-10,...Ch. 7 - Finding the Area of a RegionIn 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 RegionIn Exercises 11-14,...Ch. 7 - Finding the Area of a RegionIn Exercises 11-14,...Ch. 7 - Numerical Integration Estimate the surface area of...Ch. 7 - Revenue The models R1 = 6.4 + 0.2t + 0.01t2 and R2...Ch. 7 - Finding the Volume of a Solid In Exercises 21 and...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 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 - Length of a Catenary A cable of a suspension...Ch. 7 - Approximation Determine which value best...Ch. 7 - Surface Area Use integration to find the lateral...Ch. 7 - Surface Area The region bounded by the graphs of y...Ch. 7 - Hooke's Law A force of 5 pounds stretches a spring...Ch. 7 - Work A force of 50 pounds is needed to stretch a...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 - Boyles Law A quantity of gas with an initial...Ch. 7 - Work A windlass, 200 feet above ground level on...Ch. 7 - Work The work done by a variable force in a press...Ch. 7 - Work Find the work done by the force F shown in...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 - Centroid A blade on an industrial fan has the...Ch. 7 - Finding Volume Use the Theorem of Pappus to find...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 - Fluid Force A swimming pool is 5 feet deep at one...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 - Volume A blade is cut through the center of a...Ch. 7 - Surface Area Graph the curve 8y2=x2(1x2) Use a...Ch. 7 - Torus (a) A torus is formed by revolving the...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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