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Chapter 5 Applications of Integration

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5.1 Areas Between Curves

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5.2 and 5.3 Finding Volumes of Objects

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5.2 Volume by Slicing

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Disk Method (solid object, no hole)

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Practice example:

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Washer Method (solid object, with hole)

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Practice example:

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Both Disk and Washer methods can also be used around the y-axis!

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(Washer) (Disk)

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5.3 Volume by Shells

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For objects with or without holes:

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Add a series of “cylindrical shells”:

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More general: object not limited at the bottom by the x-axis

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Practice example:

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Example with shell about the x-axis: Two cases: a)R revolving about the x-axis b)R revolving about the line y=-2.

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SUMMARY Disk & Washer about x-axis Disk & Washer about y-axis Shell about x-axis Shell about y-axis When to use what? Answer: It will depend on the shape and symmetry of the object.

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Often, more than one method can be used, but usually one would be easier than the other(s). Example: Volume of object using WASHER then SHELL (about x-axis). Ans: V = /5

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5.4 Physical Application: Work

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Example: Object moving along a straight line with a position function s (t), The force F on the object causing it to move (in the same direction) is the product of its mass m and its acceleration: (Newton’s Second Law of Motion) Units: (in SI metric system) the mass is measured in Kilograms (kg), the displacement in Meters (m), the time in Seconds (s) the force in Newtons (N) (So that: 1 N = 1 kg m/s 2 ). the work in Joules (J) Work = total amount of effort required to perform a task Example: Work = (Force) (Distance) W = Fd

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SI Units: If F is measured in newtons and d in meters, then the unit for W is a newton-meter, which is called a joule (J). US Units: If F is measured in pounds and d in feet, then the unit for W is a foot-pound (ft-lb), which is about 1.36 J. Conversion formula: 1 ft-lb = 1.36 J Units

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Example 1: Work of a constant force: (a) How much work is done in lifting a 1.2-kg book off the floor to put it on a desk that is 0.7 m high? Use the fact that the acceleration due to gravity is g = 9.8 m/s 2. (b) How much work is done in lifting a 20-lb weight 6 ft off the ground?

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Example 1(a) – Solution The force exerted is equal and opposite to that exerted by gravity, so F = mg = (1.2)(9.8) = 11.76 N And the work done is W = Fd = (11.76)(0.7) 8.2 J

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Example 1(b) – Solution Here the force is given as F = 20 lb, so the work done is W = Fd = 20 6 = 120 ft-lb Notice that in part (b), unlike part (a), we did not have to multiply by g because we were given the weight (which is a force) and not the mass of the object. cont’d

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Work of a non constant force: The equation: W = F. d defines work as long as the force is constant What if the force is variable and depends on x? Object moves from x = a to x = b, under a force f (x), where f is a continuous function that changes with x. The Work exerted on the object is given by:

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Example 2: Work of a non constant force When a particle is located a distance x feet from the origin, a force of x 2 + 2x pounds acts on it. How much work is done in moving it from x = 1 to x = 3? Solution: ft-lb. The work done is 50/3 ft-lb.

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Work (Case of a Spring) Hooke’s Law states that the force required to maintain a spring stretched x units beyond its natural length is proportional to x: f (x) = kx where k is a positive constant called the spring constant. Hooke’s Law holds provided that x is not too large (a) Natural position of spring(b) Stretched position of spring

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Example 3: Work on a spring A force of 40 N is required to hold a spring that has been stretched from its natural length of 10 cm to a length of 15 cm. How much work is done in stretching the spring from 15 cm to 18 cm? Solution: According to Hooke’s Law, the force required to hold the spring stretched x meters beyond its natural length is f (x) = kx.

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Example 3 – Solution cont’d When the spring is stretched from 10 cm to 15 cm, the amount stretched is 5 cm = 0.05 m. This means that f (0.05) = 40, so 0.05k = 40 k = = 800 Thus f (x) = 800x and the work done in stretching the spring from 15 cm to 18 cm is = 400[(0.08) 2 – 0.05) 2 ] = 1.56 J

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5.5 Average Value of a Function

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To compute the average value of a function y = f (x), over an interval [a,b]: We start by dividing the interval [a, b] into n equal subintervals, each with length x = (b – a)/n. Then we choose points x 1 *,..., x n * in successive subintervals and calculate the average of the numbers f (x 1 *),..., f (x n *): (For example, if f represents a temperature function and n = 24, this means that we take temperature readings every hour and then average them.)

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Average Value of a Function Since x = (b – a)/n, we can write n = (b – a)/ x and the average value becomes If we let n increase, we would be computing the average value of a large number of closely spaced values.

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Average Value of a Function The limiting value is Therefore the average value of f on the interval [a, b] is

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Example Find the average value of the function f (x) = 1 + x 2 on the interval [–1, 2]. Solution: With a = –1 and b = 2 we have

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