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ESSENTIAL CALCULUS CH07 Applications of integration
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In this Chapter: 7.1 Areas between Curves 7.2 Volumes 7.3 Volumes by Cylindrical Shells 7.4 Are Length 7.5 Applications to Physics and Engineering 7.6 Differential Equations Review
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Chapter 7, 7.1, P359
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Consider the region S that lies between two curves y=f(x) and y=g(x) and between the vertical lines x=a and x=b, where f and g are continuous functions and f(x)≥g(x) for all x in [a,b]. (See Figure 1.)
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Chapter 7, 7.1, P359
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Chapter 7, 7.1, P360
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2. The area A of the region bounded by the curves y=f(x), y=g(x), and the lines x=a, x=b, where f and g are continuous and f(x)≥g(x) for all x in [a,b], is
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Chapter 7, 7.2, P365
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Chapter 7, 7.2, P366 DEFINITION OF VOLUME Let S be a solid that lies between x=a and x=b. If the cross- sectional area of S in the plane P x, through x and perpendicular to the x-axis, is A(x), where A is an integrable function, then the volume of S is
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Chapter 7, 7.2, P367
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Chapter 7, 7.2, P368
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Chapter 7, 7.2, P369
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Chapter 7, 7.2, P370
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Chapter 7, 7.2, P371
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Chapter 7, 7.2, P372
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Chapter 7, 7.3, P375
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Chapter 7, 7.3, P376 2.The volume of the solid in Figure 3, obtained by rotating about the y-axis the region under the curve y=f(x) from a to b, is Where 0≤a<b
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Chapter 7, 7.3, P377
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Chapter 7, 7.3, P378
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Chapter 7, 7.4, P380
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If we let ∆y i =y i -y i-1, then
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2. THE ARC LENGTH FORMULA If f ’ is continuous on [a,b], then the length of the curve y=f(x), a≤x≤b, is Chapter 7, 7.4, P381
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Chapter 7, 7.4, P382
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Chapter 7, 7.4, P383
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Chapter 7, 7.4, P384
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Chapter 7, 7.5, P387 Work done in moving the object from a to b
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Chapter 7, 7.6, P399 A differential equation is an equation that contains an unknown function and one or more of its derivatives. Here are some examples: 1. 2. 3.
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Chapter 7, 7.6, P400 The order of a differential equation is the order of the highest derivative that occurs in the equation. A function f is called a solution of a differential equation if the equation is satisfied when y=f(x) and its derivatives are substituted into the equation.
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Chapter 7, 7.6, P400 A separable equation is a first-order differential equation that can be written in the form
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Chapter 7, 7.6, P401 In many physical problems we need to find the particular solution that satisfies a condition of the form y(x 0 )=y 0. This is called an initial condition, and the problem of finding a solution of the differential equation that satisfies the initial condition is called an initial-value problem.
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Chapter 7, 7.6, P403 where k is a constant. Equation 7 is called the logistic differential equation
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